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the study of nuclear matter properties at high densities and at finite temperature has become an important problem since a large variety of data , where matter is being tested at extreme conditions of density , pressure and non - zero temperature , are becoming available in the modern experimental facilities which are already operational .
moreover , the constitution of the interior of neutron and protoneutron stars is also a problem which is receiving much attention in the recent literature .
the crust of the stars , where density is relatively low , can be adequately described by hadronic models .
the correct calculation of the star properties as the radius and the mass depend on the accuracy of the equation of state ( eos ) used to describe its matter .
we have checked that different models , either with constant or with density dependent meson - nucleon couplings present different features at subnuclear densities of nuclear asymmetric matter by comparing the regions of uniform unstable matter @xcite .
the parametrizations of these models generally take into account saturation properties of nuclear matter and properties of stable nuclei .
extensions of the models for very asymmetric nuclear matter or to finite temperatures show different behaviors .
another topic of great interest is the in - medium modification of vector meson properties .
it is well known that the hadron masses are much larger than the sum of its constituents .
one possible explanation for the large masses is that they may be generated dynamically @xcite .
spontaneous breaking of chiral symmetry is also related with the hadron masses . at high temperature and/or dense matter this symmetry
is expected to be at least partially restored , which modifies the hadron masses and the decay widths @xcite,@xcite .
experimental signature of the in - medium modifications of the vector mesons have been found very recently @xcite . in @xcite
the in medium modifications of the @xmath1 meson were investigated in photoproduction experiments and its mass was found to be lowered . in @xcite
the vector masses were verified to decrease by ten percent in medium .
other experimental results were also reported in the same direction @xcite and still some experiments have been proposed to detect in - medium modifications in a near future @xcite . in order to take into account the in - medium modifications of the hadrons , density dependent relativistic models are certainly more useful than models with fixed parameters . in 1991
brown and rho ( br ) @xcite proposed an in - medium scaling law for the masses and coupling constants for effective chiral lagrangians .
our proposal here is to study hadronic properties under extreme conditions using lagrangians which incorporate br scaling since this has been successfully applied to describe meson properties .
as argued in the literature the br scaling describes the behavior of the light mesons in extreme conditions very accurately .
for example the enhancement of dilepton production observed in heavy ion collisions ( s+au ) in the ceres collaboration and ( s+w ) in the helios-3 is most economically and beautifully described by a chiral lagrangian with br - scaled meson masses @xcite .
the strategy is to assume that the in - medium effective lagrangian has the same structure as in free space accordingly to the qcd constraints but with parameters which are modified in the medium .
so , using this approach we obtain an effective theory with density dependent parameters including many - body correlations in a practical framework .
it has also been shown that it is possible to obtain a relation between the effective parameters of chiral lagrangians in medium and landau fermi liquid parameters@xcite .
so , a link between the effective theory of qcd at mean field level and the many - body theory of nuclear matter is achieved . using this reasoning
the authors in @xcite proposed an effective lagrangian whose parameters scale in nuclear medium according to the brown - rho ( br ) scaling .
we have already shown that one of these density dependent models , to which we refer as tw model @xcite , originally derived at @xmath2 can be extrapolated to finite temperatures once the thermodynamical consistency remains unaltered @xcite .
the important range of temperature which is discussed lies between @xmath3 and @xmath4 mev since the liquid - gas phase transition takes place around @xmath3 mev , the phase transition from hadronic to quarkionic matter around 150 mev and the relevant temperatures in the cooling of a protoneutron star after a supernova explosion takes place go up to approximately 40 mev @xcite . in this work
we discuss another possible application of the formalism we have derived in @xcite in order to incorporate temperature effects in the study of lagrangians with br scaling .
it is well known that the same relativistic models used in order to explain data coming from heavy ion collisions at finite temperature , with appropriate parameter sets , also provide eos which can be used in the solution of the tolman - oppenheimer - volkoff differential equations @xcite for the calculation of stellar properties such as mass , radius and central energy density .
recent measurements of gravitational redshift of spectral lines provided direct constraints on the mass - to - radius ratio @xcite . in this second case , however , the interpretation of the absorption features as atomic transition lines is controversial @xcite . in recent works
@xcite we have checked that there are relativistic models which can be accommodated within these constraints .
hence , astrophysical observations can help in the choice of appropriate models to describe hadronic matter .
the most common relativistic model used in the description of hadronic matter is the non linear walecka model @xcite .
when applied in nuclear astrophysics , this model is normally extended with the inclusion of hyperons , which are expected to appear at high densities .
it was shown in @xcite that a low effective mass at saturation density makes the model inappropriate once hyperons are included .
the inclusion of hyperons makes the scalar meson interaction stronger and hence the proton and neutron effective masses decrease more rapidly with density , acquiring a negative value . as a test of the br scaled model ,
we also extend it to incorporate the eight lightest baryons and enforce @xmath0 equilibrium plus charge neutrality by accommodating the two lightest leptons as well .
the same extension is done within the tw model so that both density dependent models can be compared .
hence , the present work is organised as follows : in section ii the formalism for the br scaled model and its extension to finite temperature are presented , the results are compared with the ones obtained with other relativist models and a discussion is included . in section
iii the br scaled model is modified so that hyperons can be incorporated and @xmath0 equilibrium and charge neutrality are enforced so that an eos can be obtained and applied to compact stars . in the same section the tw model is also considered so that the results from two density dependent models are compared . in section
iv a discussion on the role played by the rearrangement term in different models is presented . finally , in the last section , the conclusions are drawn .
in its simplest version the lagrangian density reads @xcite & = & | + & & + ( _ ^-m_s^2 ( ) ^2 ) -_^+ m_v^2 ( ) _ ^ + & & -_^+^2 _ ^ , [ brlag ] .
where in the notation of @xcite @xmath5 is the nucleon field , @xmath6 the isoscalar vector field , @xmath7 an isoscalar scalar field , @xmath8 , @xmath9 is the vector isovector field , @xmath10 , @xmath11 is the isospin operator and the masses with asterisk are br - scaled as introduced in @xcite : = = = = ( ) .
[ br ] the scalings of the vector coupling constants are given by = ,   = ,   = , where @xmath12 is the nuclear saturation density .
in the original papers @xcite , @xmath13 was simply taken constant and equal to @xmath14 and hence , not density dependent and , with this simple choice a good description of the ground state was obtained . moreover ,
( ) = , [ phirho ] with @xmath15 in such a way that @xmath16 @xcite and @xmath17 and @xmath18 were taken equal or tly greater that @xmath19 . in this work
we propose another possible parametrization , based on the works @xcite , where ( ) = 1-y , [ phirho2 ] with @xmath20 in such a way that a decrease of the meson masses in medium at the saturation point is ten percent , as found experimentally @xcite .
it is also worth mentioning that , as far as we know this is the first work where asymmetry is taken into account by the appropriate inclusion of the @xmath21 meson in a br scaled model .
one would bear in mind that we are using the letter @xmath21 for both the meson and for the total baryonic density .
as those are common definitions in relativistic models we do not believe it may cause any confusion .
one can see that the lagrangian in ( [ brlag ] ) is of the form of a walecka - type lagrangian and all the finite temperature formalism that we have developed for the density dependent hadron field theory ( ddhft ) @xcite can be immediately applied to these lagrangians .
the thermodynamics of effective lagrangians with br scaling has been studied in @xcite for zero temperature .
the study of the validity of the br scaling hypothesis for the non - zero temperature case is demonstrated in what follows . from the euler - lagrange equations we obtain the field equations of motion in the mean field approximation for infinite matter , where the meson fields are replaced by their expectation values . in this approximation , the expectation value of the @xmath22 , @xmath1 and @xmath21 meson fields
are called @xmath23 , @xmath24 and @xmath25 respectively . the coupled equations
read @xmath26 \psi&=&0 , \label{dirac}\end{aligned}\ ] ] where the rearrangement term @xmath27 is given by @xmath28 and the scalar and baryonic densities are defined as @xmath29 notice that if the original parametrization for the br - scaled model is used , @xmath30 is a constant and the last term of the rearrangement vanishes . in the following discussion we consider nuclear matter in the the mean - field approximation . due to
translational and rotational invariance the lagrangian density reduces to @xmath31\psi \nonumber \\ & & -\frac{1}{2 } { m_s^*}^2 \phi_0 ^ 2 + \frac{1}{2 } { m_v^*}^2 v_{0}^2 + \frac{1}{2 } { m^*_\rho}^2 b_{0}^{2 } \label{lag2},\end{aligned}\ ] ] where @xmath32 for protons and neutrons respectively .
the conserved energy - momentum tensor can be derived in the usual fashion @xcite : @xmath33   \label{tem } .\ ] ] note that the rearrangement term included above and defined in eq.([rearr ] ) assures the energy - momentum conservation , i.e. , @xmath34 . from
the energy - momentum tensor one easily obtains the hamiltonian operator : @xmath35 where @xmath36 is identified as the effective nucleon landau mass and @xmath37 is the volume of the system .
notice that the energy density does not carry the rearrangement term because it cancels out in a mean field approximation : @xmath38 where @xmath39 and @xmath40 are the distribution functions for particles and anti - particles respectively and are calculated next . following the notation in @xcite , the thermodynamic potential can be written as @xmath41 where @xmath42 is the entropy of a classical fermi gas , @xmath43 is the temperature , @xmath44 ( @xmath45 ) is the proton ( neutron ) chemical potential and @xmath46 and @xmath47 are respectively the proton and neutron densities , calculated in such a way that @xmath48 .
we have @xmath49 where the distribution functions @xmath39 and @xmath40 for particles and anti - particles have to be derived in order to make the thermodynamic potential stationary for a system in equilibrium .
after straightforward substitutions , eq.([omega1 ] ) becomes @xmath50 for a complete demonstration of the above shown expressions obtained in a thomas - fermi approximation for the non - linear walecka model , please refer to @xcite .
at this point , eq.([omega2 ] ) is minimized in terms of the distribution functions for fixed meson fields , i.e. , @xmath51 for the particle distribution function , the above calculation yields @xmath52 where @xmath53 . a similar equation , with a sign difference is obtained for the anti - particle distribution function .
it is important to point out that the fields @xmath23 , @xmath24 and @xmath25 depend on the distribution function which appear in the definition of @xmath54 , @xmath21 and @xmath55 and hence , the whole calculation is performed self - consistently .
the effective chemical potentials are then defined as @xmath56 and the following equations for the distribution functions can be written : @xmath57}\ ; , \quad i = p , n . \label{disfun}\ ] ] in the above calculation we have used @xmath58 and @xmath59 . within the thomas - fermi approach
the pressure becomes @xmath60 it is worth mentioning that the thermodynamical consistency which requires the equality of the pressure calculated from the thermodynamical definition and from the energy - momentum tensor , discussed in @xcite , is also obeyed by the temperature dependent brown - rho scaled model .      at this point ,
the parameters used in the br method have to be fixed . through out
this paper the nucleon mass will be @xmath61 mev , the @xmath1 meson mass @xmath62 mev and the @xmath21 meson mass @xmath63 mev .
three different sets are used in @xcite . in what follows we use the parameter set called s3 in @xcite and define another one which we call ma , whose bulk properties are more similar to the nl3 @xcite parameter set but with a larger effective mass at nuclear saturation density . while s3 is a parametrization for the original br scaled model given by equation ( [ phirho ] ) , ma is a parametrization for the new scaling , given in equation ( [ phirho2 ] ) . in table
[ tab1 ] we show the s3 and ma constants and in table [ tab2 ] we display the nuclear matter bulk properties described by the different models used in this work .
it is important to point out that the value for the saturation density ( vide * in table ii ) was not given in @xcite . for the saturation density value shown in table [ tab1 ] ,
the compressibility is slightly different from what is stated in @xcite ( 260 mev ) .
this is probably a consequence of the fact that the authors in @xcite have not included the rearrangement term in their calculations .
notice that we distinguish the @xmath64 from the @xmath65 values . in the br scaled models it is the landau mass that should be identified with the nucleon effective mass determined by the qcd sum rule @xcite and its value should lie in between 0.55 m and 0.75 m.
in this work we derive a formalism to incorporate asymmetry and temperature effects in the brown - rho ( br ) scaled lagrangian model in a mean field theory . 
 the lagrangian density discussed in this work requires less parameters than the usual models with density dependent couplings . 
 we also present the formalism with the inclusion of the eight lightest baryons , two lightest leptons , @xmath0 equilibrium and charge neutrality in order to apply the br scaled model to the study of neutron stars . 
 the results are again compared with the ones obtained from another density dependent parametrization model . 
 the role played by the rearrangement term at t=0 for nuclear or neutron star matter and at finite temperature is investigated . 
 the br scaled model is shown to be a good tool in studies involving density dependent effective masses and in astrophysics applications .    
 pacs number(s):21.65.+f , 24.10.jv,26.60.+c,21.30.fe