document: the _ quark gluon plasma _ ( @xmath0 ) usually is defined as the phase of _ quantum chromodynamics _ ( @xmath2 ) 
   in which the quarks and gluons degrees of freedom , that is normally confined within the hadrons , are mostly liberated . 
   the possible phases of the @xmath2 and the precise locations of critical boundaries or points are currently being actively studied . 
   in fact , revealing the @xmath2 phase transition structure is one of the central aims of the ongoing and the future theoretical and the experimental research in the field of the hot and/or the dense @xmath2 @xcite . 
   it is about thirty years since the study of the hot and the dense nuclear matter in the form of the @xmath0 has been started . 
   the experiments at the @xmath3 s _ super proton synchrotron _ ( @xmath4 ) first tried to create the @xmath0 in the 1980s and 1990s : the first hints of the formation of a new state of matter was obtained from the @xmath4 data in terms of the global observable , the event - by - event fluctuations , the direct photons and the di - leptons . 
   the current experiments at the @xmath5 national laboratory s _ relativistic heavy ion collider _ ( @xmath6 ) are still continuing these efforts . in the april 2005 , 
   the formation of the quark matter was tentatively confirmed by the results obtained at the @xmath6 . 
   the consensus of the four @xmath6 research groups was in favor of the creation of the quark - gluon liquid at the very low viscosity @xcite . 
   since , in this new phase of matter , the quarks and the gluons are in the asymptotic freedom region , one expects that they interact weakly . 
   so , the _ perturbative _ methods can be used for such a system . 
   the asymptotic freedom suggests two procedures for the creation of the @xmath0 : @xmath7i@xmath8 the recipe for the @xmath0 at high temperature . 
   if one treats the quarks and the gluons as the massless noninteracting gas of molecules , such that the baryon density vanishes , the critical temperature which above that , the hadronic system dissolves into a system of quarks and gluons ( @xmath0 ) is @xmath9 @xcite . 
   however , the modern lattice @xmath2 calculation estimates the critical temperature , @xmath10 , to be about @xmath11 @xmath12 @xcite @xmath7ii@xmath8 the recipe for the @xmath0 at high baryon density . at zero temperature , 
   the critical baryon density required the transition to take place in @xmath13 @xcite i.e. four times the empirical nuclear matter density . on these grounds , one should expect to find the @xmath0 in two places in the nature : firstly , in the early universe , about @xmath14 after the _ cosmic big bang _ or secondly , at the core of _ super - dense stars _ such as the neutron and the quark stars . 
   this new phase of matter can also be created in the initial stage of the _ little big bang _ by means of the relativistic nucleus - nucleus collisions in the heavy ion accelerators @xcite . 
   the critical temperature ( critical baryon density ) at the zero baryon density ( zero temperature ) has been obtained simply for the non - interacting , massless up and down quarks and gluons in the reference @xcite . 
   some primary works in the zero baryon density and in the framework of the bag model have been also presented in the reference @xcite ( and the reference therein ) . for the more complicated field 
   theoretical approaches see the references @xcite and the references therein . 
   so it would be interesting to perform a similar calculation to reference @xcite , but with the non zero interaction , for the region in which both @xmath15 and @xmath16 . 
   so , in this work we generalize above calculations for the region with the finite temperature and the baryon density by considering the weakly interacting quarks in the framework of the _ one gluon exchange _ scheme . 
   since it is intended to use the @xmath17 method in our calculation , it is assumed that @xmath18 . 
   the dependence of @xmath19 on the temperature and the baryon density is ignored @xcite . 
   we perform calculations beyond the zero hadronic pressure approximation of reference @xcite and the pressure of the excluded volume hadron @xcite model is also taken into account to find the corresponding @xmath0 transition phase diagram . 
   so , the paper is organized as follows . in the section 
   @xmath20 we review the derivation of the _ one gluon exchange _ interaction formulas @xcite based on the landau fermi liquid picture @xcite . 
   the section @xmath21 is devoted to the calculation of the thermodynamics properties of the @xmath0 . 
   the result and its comparison with the strange quark matter @xcite are given in the section @xmath22 . 
   finally a conclusion and summary are presented in the section @xmath23 . 
   the landau fermi liquid theory describes the relativistic systems such as : the nuclear matter under the extreme conditions , the quark matter , the quark gluon plasma , and other relativistic plasmas . 
   the basic framework of the landau theory of relativistic fermi liquids is given by @xmath24 and @xmath25 @xcite . in this framework 
   , one can evaluate the energy density of a weakly interacting quarks in @xmath0 by the following formulas : @xmath26 with @xmath27 where @xmath28 , is the degeneracy of quarks . 
   @xmath29 , @xmath30 and @xmath31 are the kinetic and the potential energies and the familiar fermi - dirac liquid distributions of our quasi - particles , respectively . in the equation ( [ tatsumi ] ) , @xmath32 is the landau - fermi interaction function , which is a criterion for the interaction between the two quarks ( the `` @xmath33 '' superscript refers to the unpolarized quark matter @xcite ) . 
   it is related to the two - particle forward scattering amplitude i.e. : @xmath34 @xmath35 is the commonly defined lorentz invariant matrix element . 
   now , using the _ feynman rules _ for the @xmath2 , @xmath36 can be calculated . since 
   , the direct term is proportional to the trace of the _ gelman matrices _ , the color symmetric matrix element is only given by exchange contribution : @xmath37 where @xmath38 ( our choice is @xmath39 ) . since our system is unpolarized , it is possible to sum over all the spin states and get the average of this quantity as : @xmath40 then for massless up and down quarks , @xmath41 so , the total energy of interacting quarks can be evaluated by using the equations ( 1 ) , ( 2 ) , ( 3 ) and ( 7 ) . in the next section , the various thermodynamic quantities of the interacting @xmath0 like the internal energy , the pressure , the entropy , etc are calculated . 
   we begin with calculation of the free ultra - relativistic massless quarks partition function density , which follows by its non - interacting gluon contribution s . 
   since we are interested in the heavy ion processes , it is assumed that the initial state to be the nuclear matter with the equal neutron and proton densities i.e. @xmath42 . 
   so , @xmath43 and the contributions of @xmath44 and @xmath45 quarks are equal . then write the @xmath46 partition function for each quarks is written as ( @xmath47 is the volume ) , @xmath48\frac{4\pi g_{q}}{(2\pi)^{3 } } k^{2 } dk.\ ] ] this integral can be evaluated analytically i.e. , @xmath49 where @xmath50 and @xmath51 ( in the @xmath52 factor unit ) .     and the strange quark matter , respectively.,width=415 ]     and the strange quark matter , respectively.,width=415 ]     and the strange quark matter , respectively.,width=415 ]     and the strange quark matter , respectively.,width=415 ]     equation of state as a function of baryon density at two fixed temperatures . 
   the solid and dash curves are for the weakly interacting @xmath0 and the strange quark matter , respectively , but without the bag constant.,width=415 ]        .,width=415 ]     by taking into account the nuclear matter excluded volume pressure at four different bag pressures ( @xmath12).,width=415 ]    now we are in the position to evaluate the various thermodynamic quantities , such as the energy density , the entropy and the free energy from the above partition function , especially we have , @xmath53_{\cal t } & & \nonumber\\ & & \hspace{-44 mm } = \frac{4\pi g_{q}}{3(2\pi)^{3 } } ( \pi^{2 } { \cal t}^{2 } \mu+\mu^{3}),\end{aligned}\ ] ] where @xmath54 is the quark degeneracy for two , up and down quarks , flavors . 
   all of the thermodynamic quantities are obtained as a function of chemical potential ( @xmath55 ) and temperature ( @xmath56 ) . 
   the temperature is a suitable experimental quantity but the chemical potential is not . 
   so , it is better to rewrite the thermodynamic quantities as a function of the temperature and the baryon density , instead of the chemical potential . since the baryon number of a quark is @xmath57 , we have , @xmath58 , and : @xmath59 then , the chemical potential can be found as a function of @xmath60 and @xmath61 , from the above equation i.e. , @xmath62 so our thermodynamic quantities become a function of baryon density and temperature .    for the energy density of gluons we have , @xmath63 where again @xmath64 is the degeneracy of gluons . 
   our system is ultra - relativistic , so , there is a simple relation between the pressure and the energy density : @xmath65 the bag pressure and the vacuum energy contributions should be included , in addition to the quark and the gluon contribution energies , @xmath66 having the pressure , the entropy density of system is evaluated , @xmath67 to study the phases of the @xmath0 , one should concentrate on the @xmath0 and the hadronic pressures , @xmath68 and @xmath69 @xcite , respectively , i.e. for @xmath70 , the system is in the confinement phase and the quarks and the gluons are inside the bag , but for @xmath71 , the quarks and the gluons pressures can overcome to the bag and the hadronic pressures and the system is in the de - confinement phase , i.e. the @xmath0 phase is created . 
   so , the phase diagram can be extracted by solving the following equation : @xmath72 where in this work , both @xmath73 @xcite and @xmath74 @xcite cases are considered . for @xmath74 , the excluded volume effect for the nuclear matter equation of state 
   is used @xcite to calculate the hadronic pressure ( equation ( 29 ) of reference @xcite ) i.e. , @xmath75},\label{n2}\ ] ] where @xmath76 is the pressure of free ideal nucleonic ( fermion ) matter @xcite ( here , degeneracy is 4 , @xmath77 , @xmath78 and @xmath79 , e.g see the equation ( 9 ) of the reference @xcite ) . 
   now , the interaction energy between the quarks can be calculated by using the landau fermi liquid model and the results derived by us in the previous section . 
   the potential energy density of the interacting massless up or down quarks is found by using the equations ( 3 ) and ( 7 ) , @xmath80 where @xmath81 is the quark degeneracy of one flavor . by using the fermi - dirac distribution and the value of each quark flavor potential energy i.e. the equation ( 21 ) , an analytical formula for the total potential energy density of the two quark flavours 
   is written as follows : @xmath82 note that , the potential energy for massless quarks is always positive . 
   so , the interaction between quarks inside the bag is repulsive and it helps the interacting quarks and the gluons to penetrate from the bag more easily , rather than the noninteracting case , and further more , the _ 
   one gluon exchange _ interaction , because of its repulsive properties , makes the conditions easier for the system to make the transition to the @xmath0 phase . 
   the internal energy density of the @xmath0 is evaluated by performing the summation over the interacting and the non - interacting parts of the energy density of quarks , the vacuum energy and the gluon energy density which were calculated before , and having that , the other thermodynamic quantities of the @xmath0 are found . 
   as it was pointed out before , it is assumed that the bag ( hadronic ) pressure is the one has been used in the reference @xcite i.e. @xmath83@xmath84 ( zero ) , in order to compare our phase diagram with this reference , and since it is intended to compare our results with the reference @xcite , the @xmath2 coupling constant os chosen to be @xmath85 . on the other hand , for the none zero hadronic pressure 
   , the baryonic chemical potential and the bag pressure are varied to reach to the critical temperature of the lattice @xmath2 predictions @xcite , i.e. @xmath86 @xmath12 . 
   we begin by presenting the calculated free energy per baryon for both the @xmath0 and the strange quark matter @xcite in the figure 1 ( 2 ) , as a function of baryon density ( temperature ) at the two different temperatures ( baryon densities ) . the free energy for the @xmath0 is larger than those of strange quark matter , since we know that the strange quark matter should be more stable than the @xmath0 and therefore the strange quark matter free energy should be smaller than that of the @xmath0 . as one should expect , the free energy increases ( decreases ) by increasing the baryon density ( temperature ) . while , the @xmath0 has less temperature dependent with respect to the strange quark matter , they have similar density dependent at fixed temperature . 
   similar comparisons are made for the entropies in the figures 3 and 4 . 
   the entropy per baryon for the @xmath0 is an increasing ( decreasing ) function of temperature ( baryon density ) and it is smaller than that of the strange quark matter @xcite . 
   again their dependence on the density is the same , but they behave especially differently at larger temperatures .    the plots of the equation of states of both the @xmath0 ( without the effect of constant bag pressure ) and the strange quark matter as a function of baryon density at the two different temperatures are given in the figure 5 . 
   the @xmath0 equation of state is much harder than that of strange quark matter at the same baryon density and temperature . 
   the pressure of weakly interacting @xmath0 for the two different @xmath2 coupling constants , and the noninteracting @xmath0 ( without the effects of constant bag pressure ) at zero temperature as a function of baryon density is shown in the figure 6 . 
   the increase in the interaction strength makes the pressure to rise , and therefore at the smaller baryon densities , the pressure of quarks becomes equal to the bag pressure . 
   so , the interaction facilitates the quarks transition to the deconfined phase at lower density . 
   the @xmath2 coupling constant also plays the same role , i.e it will reduce the transition density .    finally , the phase diagrams for both the interacting and the noninteracting @xmath0 are shown in the figure 7 for @xmath87 . 
   the _ one gluon exchange _ interaction which is repulsive , causes to get the @xmath0 at the smaller baryon densities and temperatures . 
   as it was pointed out before , the reason is very simple , the repulsive interaction between quarks helps them to escape from the bags . 
   so the formation of the @xmath0 happens much easier for the interacting quarks than the noninteracting one . 
   but , the critical temperature is about @xmath88 , which is much less than the lattice @xmath2 suggestion of @xmath89 . in the figure 8 
   , the hadronic pressure has been also taken into the account ( see the equations ( [ n1 ] ) and the reference [ n2 ] ) . 
   the slashed area is the forbidden region , i.e. for the bag pressure approximately larger than @xmath90 there is no critical temperature with the zero baryonic chemical potential ( density ) . with the bag pressure about @xmath91 ( 
   note that the bag pressure estimated to as large as @xmath92 @xmath12 @xcite ) and the low baryonic chemical potential ( density ) it is possible to get results near to the lattice @xmath2 prediction , i.e. @xmath86 @xmath12 . 
   in conclusion , the _ one gluon exchange _ interaction was used to evaluate the strength of potential energy of the @xmath0 in the fermi liquid model . by calculating the @xmath0 partition function , the different thermodynamic properties of the @xmath0 as a function of baryon density and temperature for the both interacting and the non - interacting cases were discussed . 
   it was found that the @xmath0 internal and free energies are much larger than those of strange quark matter . on the other side , 
   if we consider the massive quarks like the strange quarks in our the @xmath0 , the potential energy becomes a negative quantity , but for the massless quarks it is always a positive quantity , therefore the internal and the free energy densities of strange quark matter become smaller than the @xmath0 ones . 
   we have seen how the _ one gluon exchange _ interaction for the massless quarks affects the phase diagram of the @xmath0 and causes the system to reach to the deconfined phase at the smaller baryon densities and temperatures . 
   our results depend on the values of bag pressure and the @xmath2 coupling constant which the latter does not have a dramatic effect on our results . 
   the increase of the hadronic and the bag pressure can improve our results toward the lattice @xmath2 calculations . in 
   the future works , we could adjust our phase diagram to get the relation between the bag constant and the @xmath2 coupling constant . on the other hand it is possible to generalize our method for the non - constant @xmath2 coupling and the bag pressure . 
   finally , we can also add the interaction between the gluons to our present calculations .
summary: the thermodynamic properties of the _ quark gluon plasma _ ( @xmath0 ) as well as its phase diagram are calculated as a function of baryon density ( chemical potential ) and temperature . 
   the @xmath0 is assumed to be composed of the light quarks only , i.e. the up and the down quarks , which interact weakly and the gluons which are treated as they are free . 
   the interaction between quarks is considered in the framework of the _ one gluon exchange _ model which is obtained from the fermi liquid picture . 
   the bag model is used , with fixed bag pressure ( @xmath1 ) for the _ nonperturbative _ part and the _ quantum chromodynamics _ ( @xmath2 ) coupling is assumed to be constant i.e. no dependence on the temperature or the baryon density . 
   the effect of weakly interacting quarks on the @xmath0 phase diagram are shown and discussed . 
   it is demonstrated that the _ one gluon exchange _ 
   interaction for the massless quarks has considerable effect on the @xmath0 phase diagram and it causes the system to reach to the confined phase at the smaller baryon densities and temperatures . the pressure of excluded volume hadron gas model is also used to find the transition phase - diagram . 
   our results depend on the values of bag pressure and the @xmath2 coupling constant which the latter does not have a dramatic effect on our calculations . finally , we compare our results with the thermodynamic properties of strange quark matter and the lattice @xmath2 prediction for the @xmath0 transition critical temperature .