The baryon mass calculation in the chiral soliton model at finite temperature and density
aa r X i v : . [ h e p - ph ] A p r The baryon mass calculation in the chiral soliton model at finite temperature anddensity
Hui Zhang, Renda Dong and Song Shu ∗ Department of Physics and Electronic Technology, Hubei University, Wuhan 430062, China
In the mean-field approximation, we have studied the soliton which is embedded in a thermalmedium within the chiral soliton model. The energy of the soliton or the baryon mass in thethermal medium has been carefully evaluated, in which we emphasize that the thermal effectivepotential in the soliton energy should be properly treated in order to derive a finite and well-definedbaryon mass out of the thermal background. The result of the baryon mass at finite temperaturesand densities in chiral soliton model are clearly presented.
PACS numbers: 12.39.Fe, 12.39.-x, 14.20.-c, 11.10.Wx
I. INTRODUCTION
Quantum chromodynamics (QCD), as the fundamental theory of strong interactions, gains an increasing number ofapplications in particle physics. Perturbative QCD calculations (lattice QCD) methods work well for small distancesor high energies. However, the analytical as well as the numerical methods have not been developed enough on thescale of large distances or low energies due to its essential non-perturbative features of hidden (spontaneously broken)chiral symmetry and confinement, especially if the baryons are involved [1–7]. Therefore, effective quark models whichhave the certain essential features of QCD are other important choices [8]. The chiral soliton model which is also calledthe linear sigma model (L σ M) [9], as one of effective models, incorporates the chiral symmetry and its spontaneousbreaking [10–14]. The model which has been proposed as a model for strong nuclear interactions provides a gooddescription of the nuclear properties. The chiral soliton model could be solved in the mean-field approximation. Asemiclassical soliton solution is referred to a baryon, and the baryon properties can be derived easily from the solitonsolution [15–20].In recent years, studies on the behavior of strongly interacting mater under extreme conditions which is createdby relativistic heavy ion collision are more and more interesting. The hadron properties (masses, radius, magneticmoments, etc.) in a hot and dense medium have drawn a lot of attentions [21–23]. The hadron properties reflect thenon-perturbative features of QCD. At zero temperature, there are sufficient studies on the hadron properties, especiallythe hadron mass, within effective models. At finite temperature, the quarks (and thus hadrons) are expected to becomelighter with chiral symmetry getting restored in the famous Brown-Rho paper [24]. This meets quite many papers,such as references [25–27]. However, there are also many papers which obtain that the hadron mass increase with thetemperature increasing [28–30].The chiral soliton model successfully predicts the static nucleon properties at zero temperature and density [15].The nucleon properties has also been studied through the same model at finite temperatures and densities [28–33].The effective masses have been obtained by these solitons. In these studies, however, the thermal medium contributionto the soliton energy is not properly treated. In reference [31–33], the energy of the thermal medium is completelyneglected, but under this treatment the baryon mass could not go back to the right result of the baryon mass at zerotemperature and density. In references [28–30], the thermal medium contribution is not well subtracted which resultsin unphysical rising of a baryon mass at high temperatures. Our goal in this paper is to give a well defined baryonmass calculation in a thermal medium within the soliton model.The structure of this paper is as follows: In section II, the chiral soliton model is introduced at zero as well asfinite temperature and density. In section III, the baryon mass calculation in medium through chiral soliton modelis discussed. In section IV, we present the thermal effective potential density, and show the soliton solutions of thechiral soliton equations at different temperatures and densities. Then the numerical results are discussed before thesummary section. ∗ Corresponding author.E-mail address:[email protected]
II. THE CHIRAL SOLITON MODEL AT FINITE TEMPERATURE AND DENSITY
The Lagrangian density of the chiral soliton model with the interactions of quarks and mesons is [15] L = ¯ ψ [ iγ µ ∂ µ + g ( σ + iγ ~τ · ~π )] ψ + 12 ( ∂ µ σ∂ µ σ + ∂ µ ~π∂ µ ~π ) − U ( σ, ~π ) , (1)where ψ represents the spin- two flavors light quark fields ψ = ( u, d ), σ is the spin-0 isosinglet scalar field, and ~π isthe spin-0 isovector pion field ~π = ( π , π , π ). The potential for σ and ~π is U ( σ, ~π ) = λ σ + ~π − ν ) + Hσ − m π λ + f π m π , (2)where the last two constant terms in equation (2) are used to guarantee that the energy of a vacuum in the absenceof quarks is zero. The minimum energy occurs for chiral fields σ and ~π restricted to the chiral circle σ + ~π = f π , (3)where f π = 93 M eV is the pion decay constant, Hσ is the explicit chiral symmetry breaking term, H = f π m π , and m π = 138 M eV being the pion mass. The chiral symmetry is explicitly broken in vacuum and the expectation valuesof the meson fields are: h σ i = − f π and h ~π i = 0. The constituent quark mass in vacuum is M q = gf π , and the σ mass is defined by m σ = m π + 2 λf π . The quantity ν can be expressed as ν = f π − m π /λ . In our calculation wefollow the choice of the reference [15] and set the constituent quark mass and the sigma mass as M q = 500 M eV and m σ = 1200 M eV that determine the parameters g ≈ .
28 and λ ≈ . T and chemical potential µ . First, wederive the thermal effective potential of the spatially uniform system at finite temperature and density using the finitetemperature field theory [34]. Ω( σ, π ; T, µ ) = U ( σ, π ) + Ω ¯ ψψ + B ( T, µ ) , (4)where B ( T, µ ) is used to guarantee that the absolute minimum value of the thermal effective potential is zero, andit is the key of strictly calculating the baryon mass, which will be discussed later. The thermodynamical potentialwhich is distributed by the homogeneous medium isΩ ¯ ψψ = − ν q T Z d ~p (2 π ) { ln[1 + e − ( E q − µ ) /T ] + ln[1 + e − ( E q + µ ) /T ] } , (5)where ν q is the degeneracy factor ν q = 2( spin ) × f lavor ) × color ) = 12 and E q = q ~p + M q is the valence quarkand antiquark energy for u , d quarks. The constituent quark (antiquark) mass M q is defined by M q = g ( σ + π ) . (6)At large radius r, the σ field assumes its vacuum value σ v . The minimum energy either in a vacuum or in a thermalvacuum for chiral fields is restricted to the chiral circle σ + π = σ v , (7)where the value of σ v in the thermal medium should be determined by the absolute minimum of the thermodynamicalpotential, which is ∂ Ω ∂σ = 0 [35].Now we embed a soliton in a homogeneous hot and dense quark medium with temperature T and chemical potential µ . Thus the effective Lagrangian is L eff = ¯ ψ [ iγ µ ∂ µ + g ( σ + iγ ~τ · ~π )] ψ + 12 ( ∂ µ σ∂ µ σ + ∂ µ ~π∂ µ ~π ) − Ω( σ, π ; T, µ ) , (8)The lagrangian can also be found in references [28, 31–33]. From the Lagrangian, the field radial equations at finitetemperature and density could be derived d u ( r )d r = − ( ǫ − gσ ( r )) v ( r ) − gπ ( r ) u ( r ) , (9)d v ( r )d r = − ( 2 r − gπ ( r )) v ( r ) + ( ǫ + gσ ( r )) u ( r ) , (10)d σ ( r )d r + 2 r d σ ( r )d r + N g ( u ( r ) − v ( r )) = ∂ Ω ∂σ , (11)d π ( r )d r + 2 r d π ( r )d r − π ( r ) r + 2 N gu ( r ) v ( r ) = ∂ Ω ∂π . (12)where ∂ Ω ∂σ = ∂U ( σ, π ) ∂σ + g σν q Z d p (2 π ) E q ( 11 + e ( E q − µ ) /T + 11 + e ( E q + µ ) /T ) , (13) ∂ Ω ∂π = ∂U ( σ, π ) ∂π + g ~πν q Z d p (2 π ) E q ( 11 + e ( E q − µ ) /T + 11 + e ( E q + µ ) /T ) . (14)In the above derivations one takes the mean-field approximation and the “hedgehog” ansatz witch means h σ ( ~r, t ) i = σ ( r ) , h ~π ( ~r, t ) i = ~rπ ( r ) , (15) ψ ( ~r, t ) = e − iǫt N X i =1 q i ( ~r ) , q ( ~r ) = (cid:18) u ( r ) i~σ · ~rv ( r ) (cid:19) χ, (16)( ~σ + ~τ ) χ = 0 , (17)where q i are N identical valence quarks in the lowest s-wave level with (eigen) energy ǫ . N is set to 3 for baryons and2 for mesons. χ is the spinor. The quark functions should satisfy the normalization condition4 π Z r ( u ( r ) + v ( r ))d r = 1 . (18)And the boundary conditions are v (0) = 0 , d σ (0)d r = 0 , π (0) = 0 , (19) u ( ∞ ) = 0 , σ ( ∞ ) = σ v , π ( ∞ ) = 0 . (20) III. THE BARYON MASS CALCULATION IN MEDIUM
At certain values of temperature and density, the equations (9)-(12) together with normalization condition (18) andboundary conditions (19),(20) which are nonlinear ordinary differential equations could be numerically solved. Usingthis solution, the physical properties of the three-quark system can be calculated. The total energy or mass of thehedgehog baryon is given by: E = M B = N ǫ + 4 π Z ∞ d r E , (21)where N is set to 3 for baryon and ǫ is the quark energy. At zero temperature and density its value is 30.5MeV. Whenthe soliton equations are solved at finite temperatures and densities, it will change with the temperature and density,which will be presented in the next section. E ( r ; T, µ ) in equation (21) is radial energy density of the meson fields, and it reads E = r [ 12 ( d σ d r ) + 12 ( d π d r ) + π r + Ω( σ, π ; T, µ )] , (22)where Ω is the thermal effective potential. This potential energy plays dual roles in the system. Assume r as theradius of a soliton. Inside the domain of the soliton or r ∼ r , Ω is the effective potential energy of the meson fields.While outside the domain of the soliton or r ≫ r , Ω is the thermodynamic potential energy of the homogeneousmedium. Since it includes the energy of the thermal background, it should be properly subtracted off. Otherwisethe integral of the soliton energy would be infinite. In some previous studies [31–33] this energy has been completelyneglected, in order to make the integral finite. However, this treatment is not proper, as the soliton energy could notgo back to the right form when temperature and density go to zero. In other studies [28–30], the background energyhas not been well subtracted, and the baryon mass becomes unphysically large at high temperatures or densities asa result.Now, let us make an analysis of the integral of the soliton energy. At zero temperature and density, the thermaleffective potential Ω becomes to the potential U ( σ, π ). When r → ∞ , the meson fields σ and π assume their vacuumvalues, which are determined by ∂U ( σ,π ) ∂σ = ∂U ( σ,π ) ∂π = 0. At these values, the potential U ( σ, π ) has the lowest minimumenergy which is zero corresponding to the vacuum. From (21) one could see that the integral is finite as r → ∞ , U ( σ, π ) →
0. Now let us take a look of the case at finite temperature and densitiy. U ( σ, π ) is replaced by Ω( σ, π ). When r → ∞ , the meson fields σ and π assume their thermal vacuum values, which are determined by ∂ Ω( σ,π ) ∂σ = ∂ Ω( σ,π ) ∂π = 0.At these values, the thermal effective potential Ω( σ, π ) has the lowest minimum energy but nonzero which representsthe energy of the thermal vacuum. From (21) one could see that the integral is infinite because Ω( σ, π ) approaches anonzero value as r → ∞ . The thermal background energy has been included in evaluating the soliton energy, thereforeit is infinite.How to properly subtracted this background energy? We think that one should make a redefinition of the energyof the thermal vacuum when evaluating the integral, that is to say, when r → ∞ , one should set Ω( σ, π ) →
0. Thiscould be fulfilled by readjust the B ( T, µ ) to make the minimum of thermodynamic potential Ω always staying at zeroat different temperatures and densities. By this treatment, the integral becomes finite, and the energy of the thermalbackground has been successfully subtracted off. Thus we obtain a finite and well defined baryon mass in thermalmedium.
IV. NUMERICAL RESULTS - - - Σ H fm L W H f m - L - - - Σ H fm L W H f m - L (a) (b)FIG. 1: The thermodynamical potential Ω, (a) at µ = 0, T = { , , } MeV . (b) at µ = 300 MeV , T = { , , } MeV . It is instructive to plot the thermal effective potential as a function of the σ filed for different temperatures anddensities. In Fig.1, the left part (a) is for µ = 0 and the right (b) for µ = 300 M eV . One can see that by addingthe temperature and density dependent parameter B ( T, µ ) we have shifted the minimum value of thermal effectivepotential to zero for different temperatures and densities. This means when σ = σ v at different temperatures anddensities we always have Ω = 0. As a result, the thermal background energy has been subtracted out of the solitonenergy as in (21). In our subtraction approach, B ( T, µ ) plays an important role in deriving the finite baryon mass.At different temperatures and densities, the numerical results of B ( T, µ ) are shown in Table.I. It is sensitive to thevariation of the temperature, but not of the chemical potential. B ( T, µ ) increases with the temperature and chemicalpotential increasing.In Fig.2, we plot the quark fields u ( r ), v ( r ) and the meson fields σ ( r ), π ( r ) as functions of the radius r at fixedchemical potential for different temperatures: the left (a) is for µ = 0 and the right (b) for µ = 300 M eV . It can beseen that in both cases the amplitudes of the soliton solutions decrease and change more and more rapidly with thetemperature increasing. Μ= = = = - H fm L u , v H f m - (cid:144) L Σ , Π H f m - L u v Σ Π Μ=
300 MeVT = =
100 MeVT = - H fm L u , v H f m - (cid:144) L Σ , Π H f m - L u v Σ Π (a) (b)FIG. 2: The quark fields u ( r ), v ( r ) and the meson fields σ ( r ), π ( r ) as functions of the radius r , (a) at µ = 0 MeV , T = { , , } MeV . (b) at µ = 300 MeV , T = { , , } MeV .TABLE I: The energy or mass M B of baryon, eigenvalue ǫ and B ( T, µ ) at different temperatures and chemical potentials. µ = 0 T(MeV) 0 170 185 ǫ ( fm − ) 0.216 0.289 0.451 B ( fm − ) 0 0.439 0.746 µ = 300MeV T(MeV) 0 100 110 ǫ ( fm − ) 0.216 0.257 0.288 B ( fm − ) 0 0.074 0.202 In solving u and v fields, one should notice that the normalization condition (18) must be observed at differenttemperature and densities. This makes quark energy ǫ changing with temperature and density, which is shown inTable.I. ǫ increases with the temperature and chemical potential increasing. From σ and π together with thermaleffective potential Ω in which the thermal background energy has been subtracted off, one can obtain the energydensity E by (22). Μ= = = = H fm L E H f m - L Μ= = = = H fm L E H f m - L (a) (b)FIG. 3: The energy density E ( r ) as functions of the radius r , (a) at µ = 0, T = { , , } MeV . (b) at µ = 300 MeV , T = { , , } MeV . In Fig.3, the energy density E are plotted as functions of r at different temperatures for µ = 0 and µ = 300 M eV .One can see that as r → ∞ , we have E →
0. This can only be fulfilled when the thermal background energy is wellsubtracted off. One can also see that the amplitude of E decreases more and more rapidly, and the width of E gets“fatter” with the temperature increasing. The position of the peak value which represents the radius of the solitonincreases with the temperature increasing. It means that the spatially localized energy distribution expands withthe temperature increasing. One can see from Fig.3, the area under the curve of function E ( r ) diminishes with thetemperature increasing, which results in that the meson energy decreases with the temperature increasing.From Eq.(21,22), one could see that the soliton energy comes from the summation of the quark energy and themeson energy. Although the quark energy is increasing with temperature or chemical potential increasing, the decreaseof the meson energy outweighs the increase of quark energy. As a result the total energy or the baryon mass willdecrease with temperature or chemical potential increasing, which is presented in Fig.4. And the baryon mass E B decreases more and more rapidly with the temperature or chemical potential increasing. Other works employingthe same model had presented the baryon mass [28–31]. Our curve is quite similar with the curve E ∗ of Fig.4 inRef.[31]. However, they treated the effective potential in the soliton energy as the background medium attribution,and completely neglected it, which also makes the baryon mass finite. From our discussion here, it could be seen thistreatment of subtracting the background energy is not proper. In their results, the rate of decline of baryon masswith temperature increasing is larger than ours. In references [28–30], the authors had not subtracted the thermalbackground energy from the thermodynamic potential. Therefore, in their results the baryon mass is infinite, but theyhad just made a cut-off. This scheme made the baryon mass unphysically large at high temperatures or densities.In references [32, 33], the authors had obtained the nucleon properties through the soliton in the NJL model. Thenucleon mass decreases with temperature increasing at zero density, while at finite density it increases at first andthen decreases with temperature increasing. At high temperatures, the qualitative results of the decreasing of thenucleon mass is consistent with ours and those in Ref.[31]. Μ= Μ= Μ= H MeV L E B H M e V L T = = = Μ H MeV L E B H M e V L (a) (b)FIG. 4: The Baryon mass E B of a stable chiral soliton (a) as a function of the temperature T at µ = 0, µ = 200 MeV and µ = 300 MeV . (b) as a function of the temperature µ at T = 50 MeV , T = 100 MeV and T = 150 MeV . V. SUMMARY
In this paper, we have studied the chiral soliton model at finite temperature and density, and solved the chiralsoliton equations at different temperatures and densities with different boundary conditions. By properly subtract-ing the thermal background energy we have obtained a strictly well-defined finite baryon mass in soliton model atfinite temperature and density. As a result the baryon mass decreases with the temperature and chemical potentialincreasing, which is consistent with the Brown-Rho scaling.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China with No. 10905018 and No.11275082. [1] D.H. Rischke, Prog. Part. Nucl. Phys. 52 (2004) 197.[2] K. Yagi, T. Hatsuda and Y. Miake, ”Quark-gluon plasma: From big bang to little bang,” Camb.Monogr. Part. Phys. Nucl.Phys. Cosmol. 23 (2005) 1.[1] D.H. Rischke, Prog. Part. Nucl. Phys. 52 (2004) 197.[2] K. Yagi, T. Hatsuda and Y. Miake, ”Quark-gluon plasma: From big bang to little bang,” Camb.Monogr. Part. Phys. Nucl.Phys. Cosmol. 23 (2005) 1.