The Surface Tension of Magnetized Quark Matter
aa r X i v : . [ h e p - ph ] J un The Surface Tension of Magnetized Quark Matter
Andr´e F. Garcia and Marcus Benghi Pinto ∗ Departamento de F´ısica, Universidade Federal de Santa Catarina, 88040-900 Florian´opolis, Santa Catarina, Brazil (Dated: September 24, 2018)The surface tension of quark matter plays a crucial role for the possibility of quark matter nucle-ation during the formation of compact stellar objects and also for the existence of a mixed phasewithin hybrid stars. However, despite its importance, this quantity does not have a well establishednumerical value. Some early estimates have predicted that, at zero temperature, the value fallswithin the wide range γ ≈ −
300 MeV / fm but, very recently, different model applications havereduced these numerical values to fall within the range γ ≈ −
30 MeV / fm which would favor thephase conversion process as well as the appearance of a mixed phase in hybrid stars. In magnetarsone should also account for the presence of very high magnetic fields which may reach up to about eB ≈ − m π ( B ≈ − G ) at the core of the star so that it may also be important toanalyze how the presence of a magnetic field affects the surface tension. With this aim we considermagnetized two flavor quark matter, described by the Nambu–Jona-Lasinio model. We show thatalthough the surface tension oscillates around its B = 0 value, when 0 < eB < ∼ m π , it only reachesvalues which are still relatively small. For eB ≈ m π the B = 0 surface tension value drops by about30% while for eB > ∼ m π it quickly raises with the field intensity so that the phase conversion andthe presence of a mixed phase should be suppressed if extremely high fields are present. We alsoinvestigate how thermal effects influence the surface tension for magnetized quark matter. PACS numbers: 21.65.Qr., 26.60.Kp, 11.10.Wx, 11.30.Rd, 12.39.Ki
I. INTRODUCTION
The understanding of compact stars requires the study of strongly interacting matter at low temperatures and highchemical potentials. However, this portion of the QCD phase diagram cannot be addressed by current lattice-QCDmethods so that studies of this phase region must rely on less fundamental models. Most investigations suggest thatthere is a first-order chiral phase transition which, for T ≈
0, sets in at baryon densities several times that of thenuclear saturation density, ρ ≈ .
17 fm − . The expected phase transition will have significant implications for thepossible existence of quark stars and the possibilities depend on the dynamics of the phase conversion as well as onthe time scales involved [1–5]. When the phase diagram of bulk matter exhibits a first-order phase transition, the twophases, associated with a high and a low density value ( ρ H and ρ L ), may coexist in mutual thermodynamic equilibriumand, consequently, when brought into physical contact a mechanically stable interface will develop between them. Theassociated surface tension γ T depends on the temperature T ; it has its largest magnitude at T = 0 approaching zeroas T is increased to the critical end point temperature, T c , where the first order transition line terminates. Thesurface tension plays a key role in the phase conversion process and it is related to various characteristic quantitiessuch as the nucleation rate, the critical bubble radius, and the favored scale of the blobs generated by the spinodalinstabilities [6, 7]. For our present purposes it is important to remark that a small surface tension would facilitatevarious structures in compact stars, including the presence of mixed phases in a hybrid star [8].Unfortunately, despite its central importance, the surface tension of quark matter is rather poorly known. Atvanishing temperatures, some early estimates fall within a wide range, typically γ ≈ −
50 MeV / fm [9, 10] andvalues of γ ≈
30 MeV / fm have been considered for studying the effect of quark matter nucleation on the evolutionof proto-neutron stars [11]. But the authors in Ref. [12], taking into account the effects from charge screening andstructured mixed phases, estimate γ ≈ −
150 MeV / fm , without excluding smaller values, and even a higher value, γ ≈
300 MeV / fm , was found on the basis of dimensional analysis of the minimal interface between a color-flavorlocked phase and nuclear matter [13].More recently, the surface tension for two-flavor quark matter was evaluated, in Ref. [14], within the the quarkmeson model (QM), in the framework of the thin-wall approximation for bubble nucleation . The predicted valuescover the 5 −
15 MeV / fm range, depending on the inclusion of vacuum and/or thermal corrections. In principle, thisrange makes nucleation of quark matter possible during the early post-bounce stage of core-collapse supernovae and ∗ Electronic address: [email protected] it is thus a rather important result.The Nambu–Jona-Lasinio model (NJL) with two and three flavors was subsequently considered in the evaluationof γ T [15] via a geometrical approach introduced by Randrup in Ref. [6]. This method makes it possible to expressthe surface tension for any subcritical temperature in terms of the free energy density for uniform matter in theunstable density range. In practice, the procedure is rather simple to implement and it provides an estimate forthe surface tension that is consistent with the equation of state (EoS) implied by the adopted model, with itsspecific approximations and parametrizations. The results obtained in Ref. [15] predict that, at zero temperature, γ ≈ −
30 MeV / fm depending on the chosen parameters.Very recently, the Polyakov quark meson model (PQM) with three flavors has been considered [16] in the contextof the thin wall approximation extending the work of Ref. [14] with confinement and strangeness. Depending on theadopted parametrization, the numerical results obtained in Ref. [16] are within the γ ≈ −
28 MeV / fm range.The authors have confirmed that the inclusion of the strange sector, which was originally done in Ref. [15], does notchange appreciably the dynamics of the transition at low temperatures and high chemical potentials as neither doesthe inclusion of the Polyakov loop. Regarding the possibility of phase conversion taking place, within compact stellarobjects, it is important to remark that all these three recent evaluations [14–16] predict values for the surface tensionwhich are low enough so that, in principle, the phase conversion phenomenon could take place. At the sam e time,these three estimates, of low values, favor the appearance of a mixed phase within a hybrid star.One should also recall that very high magnetic fields can be present in magnetars reaching up to eB ≈ − m π ( B ≈ − G ), or higher, at the core of the star [17]. In many applications this type of compact stellar objectsare modeled as a hybrid star which has a core of quark matter surrounded by hadronic matter [18] and if the surfacetension between the two phases is small enough, as predicted by Refs. [14–16], the transition occurs via a mixedphase (Gibbs construction). On the other hand, if γ T has a high value, as predicted by Refs. [12, 13], it occursat a sharp interface (Maxwell construction) [19]. Therefore, the value of the surface tension in the presence of highmagnetic fields may be an important ingredient for investigations related to quark and hybrid stars. Since this typeof evaluation does not seem to have been carried out before we intend to perform such a calculation here by extendingthe work of Ref. [15] so as to account for the presenc e of high magnetic fields. The coexistence region associatedwith the first order transition of strongly interacting magnetized matter has been recently investigated in Ref. [20]which predicts, as one of its main results, that the value of ρ H oscillates around the B = 0 value for 0 < eB < ∼ m π and then grows for higher values. Taking into account that γ T depends on the difference between ρ H and ρ L [6, 7]one may then expect to find a similar behavior here. Indeed, as we will demonstrate, when a magnetic field is presentthe surface tension value oscillates very mildly for 0 < eB < ∼ m π before decreasing in a significant way between4 m π < ∼ eB < ∼ m π . Then, after reaching a minimum at eB ≈ m π it starts to increase continuously reaching the B = 0 value at eB ≈ m π which allows to conclude that the existence of a mixed phase remains possible within thisrange of magnetic fields. For eB values higher than ≈ m π this quantity increases rapidly with the magnetic fielddisfavoring the presence of a mixed phase within hybrid stars. We also show how the temperature affects γ T ( B ) bydecreasing its value towards zero which is achieved at T = T c , as already emphasized.The paper is organized as follows. In the next section we review the method for extracting the surface tension fromthe equation of state. In Sec. III we present the EoS for the magnetized two flavor NJL. Then, in Sec. IV, we presentour numerical results. The conclusions and final remarks are presented in Sec. V. II. THE GEOMETRIC APPROACH TO THE SURFACE TENSION EVALUATION
To make this work self contained let us review, in this section, the geometric approach to the surface tensionevaluation which was originally proposed in Ref. [6]. We first assume that the material at hand, strongly interactingmatter, may appear in two different phases under the same thermodynamic conditions of temperature T , chemicalpotential µ , and pressure P . These two coexisting phases have different values of other relevant quantities, such asthe energy density E , the net quark number density ρ , and the entropy density s . Under such circumstances, the twophases will develop a mechanically stable interface if placed in physical contact. An interface tension, γ T , is thenassociated to this interface.The two-phase feature appears for all temperatures below the critical value, T c . Thus, for any subcritical temper-ature, T < T c , hadronic matter at the density ρ L ( T ) has the same chemical potential and pressure as quark matterat the (larger) density ρ H ( T ). As T is increased from zero to T c , the coexistence phase points ( ρ L , T ) and ( ρ H , T )trace out the lower and higher branches of the phase coexistence boundary, respectively, gradually approaching eachother and finally coinciding for T = T c . Any ( ρ, T ) phase point outside of this boundary corresponds to thermody-namically stable uniform matter, whereas uniform matter prepared with a density and temperature corresponding toa phase point inside the phase coexistence boundary is thermodynamically unstable and prefers to separate into twocoexisting thermodynamically stable phases separated by a mechanically stable interface. Because such a two-phaseconfiguration is in global thermodynamic equilibrium, the local values of T , µ , and P remain unchanged as one movesfrom the interior of one phase through the interface region and into the interior of the partner phase, as the localdensity ρ increases steadily from the lower coexistence value ρ L to the corresponding higher coexistence value ρ H .It is convenient to work in the canonical framework where the control parameters are temperature and density.The basic thermodynamic function is thus f T ( ρ ), the free energy density as a function of the (net) quark numberdensity ρ for the specified temperature T . The chemical potential can then be recovered as µ T ( ρ ) = ∂ ρ f T ( ρ ), andthe entropy density as s T ( ρ ) = − ∂ T f T ( ρ ), so the energy density is E T ( ρ ) = f T ( ρ ) − T ∂ T f T ( ρ ), while the pressure is P T ( ρ ) = ρ∂ ρ f T ( ρ ) − f T ( ρ ).For single-phase systems f T ( ρ ) is convex, i.e. its second derivative ∂ ρ f T ( ρ ) is positive, while the appearance of aconcavity in f T ( ρ ) signals the occurrence of phase coexistence, at that temperature. This is easily understood becausewhen f T ( ρ ) has a local concave anomaly then there exist a pair of densities, ρ L and ρ H , for which the tangentsto f T ( ρ ) are common. Therefore f T ( ρ ) has the same slope at those two densities, so the corresponding chemicalpotentials are equal, µ T ( ρ L ) = ∂ ρ f T ( ρ L ) = ∂ ρ f T ( ρ H ) = µ T ( ρ H ). Furthermore, because a linear extrapolation of f T ( ρ ) leads from one of the touching points to the other, also the two pressures are equal, P T ( ρ L ) = ρ L ∂ ρ f T ( ρ L ) − f T ( ρ L ) = ρ H ∂ ρ f T ( ρ H ) − f T ( ρ H ) = P T ( ρ H ). So uniform matter at the density ρ L has the same temperature, chemicalpotential, and pressure as uniform matter at the density ρ H . The common tangent between the two coexistence pointscorresponds to the familiar Maxwell construction and shall here be denoted as f MT ( ρ ). Obviously, f T ( ρ ) and f MT ( ρ )coincide at the two coexistence densities and, furthermore, f T ( ρ ) exceeds f MT ( ρ ) for intermediate densities. Thereforewe have ∆ f T ( ρ ) ≡ f T ( ρ ) − f MT ( ρ ) ≥ T , we now consider a configuration in which the two coexisting bulk phasesare placed in physical contact along a planar interface. The associated equilibrium profile density is denoted by ρ T ( z ) where z denotes the location in the direction normal to the interface. In the diffuse interface region, thecorresponding local free energy density, f T ( z ), differs from what it would be for the corresponding Maxwell system, i.e. a mathematical mix of the two coexisting bulk phases with the mixing ratio adjusted to yield an average densityequal to the local value ρ ( z ). This local deficit amounts to δf T ( z ) = f T ( z ) − f i − f T ( ρ H ) − f T ( ρ L ) ρ H − ρ L ( ρ T ( z ) − ρ i ) , (2.1)where ρ i is either one of the two coexistence densities. The function δf T ( z ) is smooth and it tends quickly to zeroaway from the interface where ρ T ( z ) rapidly approaches ρ i and f T ( z ) rapidly approaches f T ( ρ i ). The interface tension γ T is the total deficit in free energy per unit area of planar interface, γ T = Z + ∞−∞ δf T ( z ) dz . (2.2)As discussed in Ref. [6], when a gradient term used to take account of finite-range effects, the tension associatedwith the interface between the two phases can be expressed without explicit knowledge about the profile functionsbut exclusively in terms of the equation of state for uniform (albeit unstable) matter, γ T = a Z ρ H ( T ) ρ L ( T ) [2 E g ∆ f T ( ρ )] / dρρ g , (2.3)where ρ g is a characteristic value of the density and E g is a characteristic value of the energy density, while theparameter a is an effective interaction range related to the strength of the gradient term, C = a E g / ( ρ g ) . We choosethe characteristic phase point to be in the middle of the coexistence region, ρ g = ρ c and E g = [ E ( ρ c ) + E c ] /
2, where E ( ρ c ) is energy density at ( ρ c , T = 0), while E c is energy density at the critical point ( ρ c , T c ). The length a is asomewhat adjustable parameter governing the width of the interface region and the magnitude of the tension [6]. InRef. [15] this parameter was set to a ≈ /m σ ≈ .
33 fm which, also, is approximately the value found in an applicationof the Thomas-Fermi approximation to the NJL model [21]. Therefore, we shall adopt the value a = 0 .
33 fm throughoutthe present work. With these parameters fixed (see Ref. [15]), the interface tension can be calculated once the freeenergy density f T ( ρ ) is known for uniform matter in the unstable phase region, ρ L ( T ) ≤ ρ ≤ ρ H ( T ). III. THE EOS FOR THE MAGNETIZED TWO FLAVOR NJL QUARK MODEL
The NJL model is described by a Lagrangian density for fermionic fields given by [22] L NJL = ¯ ψ ( i∂/ − m ) ψ + G (cid:2) ( ¯ ψψ ) − ( ¯ ψγ ~τ ψ ) (cid:3) , (3.1)where ψ (a sum over flavors and color degrees of freedom is implicit) represents a flavor iso-doublet ( u, d type ofquarks) N c -plet quark fields, while ~τ are isospin Pauli matrices. The Lagrangian density (3.1) is invariant under(global) U (2) f × SU ( N c ) and, when m = 0, the theory is also invariant under chiral SU (2) L × SU (2) R . Within theNJL model a sharp cut off (Λ) is generally used as an ultra violet regulator and since the model is nonrenormalizable,one has to fix Λ to a value related to the physical spectrum under investigation. This strategy turns the 3+1 NJLmodel into an effective model, where Λ is treated as a parameter. Then, the phenomenological values of quantitiessuch as the pion mass ( m π ), the pion decay constant ( f π ), and the quark condensate ( h ¯ ψψ i ) are used to fix G , Λ, and m . Here, we choose the set Λ = 590 MeV and G Λ = 2 .
435 with m = 6 MeV in order to reproduce f π = 92 . m π = 140 . h ¯ ψψ i / = − . NJL = ( M − m ) G + i Z d p (2 π ) ln[ − p + M ] , (3.2)where M is the constituent quarks mass. In order to study the effect of a magnetic field in the chiral transition atfinite temperature and chemical potential a dimensional reduction is induced via the following replacements in Eq.(3.2) [27] p → i ( ω ν − iµ ) ,p → p z + (2 n + 1 − s ) | q f | B , with s = ± , n = 0 , , ... , Z + ∞−∞ d p (2 π ) → i T | q f | B π ∞ X ν = −∞ ∞ X n =0 Z + ∞−∞ dp z π , where ω ν = (2 ν + 1) πT , with ν = 0 , ± , ± ... represents the Matsubara frequencies for fermions, n represents theLandau levels and | q f | is the absolute value of the quark electric charge ( | q u | = 2 e/ | q d | = e/ e = 1 / √ ). Note also that here we have taken the chemical equilibrium condition by setting µ u = µ d = µ . Then, following Ref. [25], we can write the free energy asΩ NJL = ( M − m ) G + Ω NJLvac + Ω
NJLmag + Ω
NJLmed , (3.3)where Ω NJLvac = − N c N f Z d p (2 π ) ( p + M ) / . (3.4)This divergent integral is regularized by a sharp cut-off, Λ, yieldingΩ NJLvac = N c N f π (cid:26) M ln (cid:20) (Λ + ǫ Λ ) M (cid:21) − ǫ Λ Λ (cid:2) Λ + ǫ (cid:3)(cid:27) , (3.5)where we have defined ǫ Λ = √ Λ + M . The magnetic and the in-medium terms are respectively given byΩ NJLmag = − N c π d X f = u ( | q f | B ) ( ζ (1 , ( − , x f ) −
12 [ x f − x f ] ln( x f ) + x f ) , (3.6) We use Gaussian natural units where 1 MeV = 1 . × G which sets m π /e ≃ × G . and Ω NJLmed = − N c π d X f = u ∞ X k =0 α k | q f | B Z + ∞−∞ dp z π n T ln[1 + e − [ E p, k ( B )+ µ ] /T ] + T ln[1 + e − [ E p, k ( B ) − µ ] /T ] o . (3.7)In the last equation we have replaced the label n by k in the Landau levels in order to account for the degeneracyfactor α k = 2 − δ k . Also, in Eq (3.6) we have used x f = M / (2 | q f | B ) and ζ (1 , ( − , x f ) = dζ ( z, x f ) /dz | z = − with ζ ( z, x f ) representing the Riemann-Hurwitz function (the details of the manipulations leading to the equations abovecan be found in the appendix of Ref. [25]). Finally, in Eq. (3.7) we have E p, k ( B ) = p p z + 2 k | q f | B + M where M is the effective self consistent quark mass M = m + N c N f M Gπ ( Λ p Λ + M − M " (Λ + √ Λ + M ) M + N c M Gπ d X f = u | q f | B (cid:26) ln[Γ( x f )] −
12 ln(2 π ) + x f −
12 (2 x f −
1) ln( x f ) (cid:27) − N c M G π d X f = u ∞ X k =0 α k | q f | B Z ∞−∞ dp z E p,k ( B ) (cid:26) e [ E p,k ( B )+ µ ] /T + 1 + 1 e [ E p,k ( B ) − µ ] /T + 1 (cid:27) . (3.8)Note that in principle one should have two coupled gap equations for the two distinct flavors: M u = m u − G ( h ¯ uu i + h ¯ dd i ) and M d = m d − G ( h ¯ dd i + h ¯ uu i ) where h ¯ uu i and h ¯ dd i represent the quark condensates which differ, due to thedifferent electric charges. However, in the two flavor case, the different condensates contribute to M u and M d in asymmetric way and since m u = m d = m one has M u = M d = M .The minimum value of the grand potential represents minus the equilibrium pressure, Ω min ( T, µ ) = − P , so the netquark number density is given by ρ = ( ∂P/∂µ ) T . The entropy density given by s = ( ∂P/∂T ) µ , while the energydensity, E , can then be obtained by means of the standard thermodynamic relation P = T s −E + µρ . The knowledge ofall these quantities allow us to determine the free energy density, f ≡ E − T s = µρ − P , as well as the numerical inputs ρ H , ρ L , ρ g , and ǫ g which are needed in the evaluation of the surface tension. As already emphasized, the numericalvalue for the length scale a is chosen to be 1 /m σ ≃ .
33 fm (which is about the value found in a Thomas-Fermiapplication to the NJL model [21]).
IV. NUMERICAL RESULTS
Let us start the numerical evaluations by obtaining the phase diagram in the T − ρ B plane in order to determinethe values of essential quantities such as T c , µ c , ρ H , ρ L which allow for the evaluation of the inputs ρ g , and E g foreach value of B . As it is well known, for given subcritical temperature in the T − ρ B plane one observes that theassociated density region is bounded by the two coexistence densities ρ L and ρ H , for which the chemical potential µ has the same value, as does the pressure P . As the density ρ is increased through the lower mechanically metastable(nucleation) region, µ and P rise steadily until the lower spinodal boundary has been reached. Then, as ρ movesthrough the mechanically unstable (spinodal) region, both µ and P decrease until the higher spinodal boundary isreached. They then increase again as ρ moves through the higher mechanically metastable (bubble-formation) region,until they finally regain their original values at ρ = ρ H . Fig. 1 displays the coexistence region, in the T − ρ B plane,for B = 0, eB = 6 m π , and eB = 15 m π . Noting that ρ H oscillates around the B = 0 value and recalling that γ T depends on the difference between ρ L and ρ H , see Eq. (2.3), one can then expect that the surface tension value at eB = 6 m π will be smaller than at B = 0, at least for small temperatures. On the contrary, for eB = 15 m π , one mayexpect γ T to assume values much larger than those obtained in the B = 0 case. These expectations will be explicitlyconfirmed by our evaluation of γ T . A. The zero temperature case
In order to illustrate how the method works and also to understand the type of oscillation displayed by Fig. 1it is convenient to concentrate in the T = 0 limit since, in this case, the momentum integrals appearing in thethermodynamical potential can be performed producing equations which are easy to be analyzed from an analyticalpoint of view. Apart from that, this limit is very often considered in evaluations of the EoS for cold stars and it willbe our starting point here. Then, in the next subsection we will analyze how the surface tension is influenced bythermal effects. At T = 0 (and also at any other subcritical temperature) the grand potential can present multipleextrema representing stable, metastable, and spinodally unstable matter in the neighborhood of the phase coexistencechemical potential and, as emphasized in Ref. [15], the extraction of the surface tension by the geometric approachrequires the consideration of all these extrema. In our case it is then important to know all the gap equation solutionsas displayed in Fig. (2) which shows the effective quark mass, at T = 0, for B = 0, eB = 6 m π , and eB = 15 m π . Thiseffective mass is then used to determine the pressure from where all the other thermodynamic quantities, includingthe density, can be derived. In this figure, the continuous lines represent the stable solutions only and determinethe Maxwell line which links the high effective mass value ( M H ) to its low value ( M L ) at the coexistence chemicalpotential where the phase transition occurs. With these two stable solutions and upon using the Maxwell constructionone obtains f MT . The dashed lines are obtained by considering the unstable as well as the metastable gap equationsolutions which lie within the spinodal region. Considering all the gap equation solutions one then obtains f T ( ρ ) todetermine the difference ∆ f T ( ρ ) which is the crucial ingredient in the surface tension evaluation. But before carryingout the evaluation let us discuss the origin of the the de Hass-van Alphen oscillations, for ρ H , which appear in Fig. 1at B = 0. Note from Fig. 2 that, at the coexistence chemical potential, the gap equation for eB = 6 m π , where theoscillations are more pronounced, presents more solutions than the case B = 0 or the case eB = 15 m π . Then, theeffective mass behavior displayed in Fig. 2 allows us to understand the ρ H oscillations, shown in Fig. 1, by reviewingthe discussion carried out in Ref. [20]. There it is shown that the decrease in ρ H for eB = 6 m π , at low temperatures,can be understood in terms of the filling of the Landau levels. With this aim, we present Fig. 3 which displays thebaryonic density and the effective quark mass as functions of the magnetic field at T = 0. To analyze the figure letus recall that, in the limit T →
0, the baryonic density can be written as [25]. àà ææ òò m Π m Π Ρ B (cid:144) Ρ T @ M e V D FIG. 1: Phase coexistence boundaries in the T − ρ B plane ( ρ B appears in units of the nuclear matter density, ρ = 0 .
17 fm − ).The solid symbols indicate the location of the critical point for each value of B which occur at ( T c = 81 . µ c = 324 . B = 0, ( T c = 84 . µ c = 324 .
7) for eB = 6 m π , and ( T c = 115 . µ c = 279 MeV) for eB = 15 m π . Taken fromRef. [20]. ρ B ( µ, B ) = θ ( k F ) d X f = u k f,max X k =0 α k | q f | BN c π k F , (4.1) There is a misprint in Eq. (30) of Ref. [25] where it should be ρ B instead of ρ . where k F = p µ − | q f | kB − M and k f,max = µ − M | q f | B , (4.2)or the nearest integer. Eq. (4.1) shows that if k F < ρ B = 0 which is precisely the low density value at T = 0 B =
250 300 350 400 4500100200300400500 Μ @ MeV D M @ M e V D eB = m Π
250 300 350 400 4500100200300400500 Μ @ MeV D M @ M e V D eB = m Π
250 300 350 400 4500100200300400500 Μ @ MeV D M @ M e V D FIG. 2: The quark effective mass, at T = 0, as a function of µ for B = 0 (left panel), eB = 6 m π (center panel), and eB = 15 m π (right panel). The continuous lines indicate the gap equation stable solutions and the dashed lines the unstable and metastableones. which is easy to understand by recalling that the effective mass is double valued when the first order transition occurspresenting a high ( M H ) and a low ( M L ) value with M L < M H for T < T c and M L = M H at T = T c . Now, at T = 0, M H corresponds to the value effective quark mass acquires when T = 0 and µ = 0 (the vacuum mass) whichcorresponds to M H ≃
403 MeV at B = 0, M H ≃
416 MeV at eB = 6 m π , and M H ≃
467 MeV at eB = 15 m π . Onthe other hand, at T = 0 the first order transition happens when µ ≃
383 MeV for B = 0, µ ≃
370 MeV for eB = 6 m π and µ ≃
339 MeV for eB = 15 m π so that ρ L = 0 even at the lowest Landau level (LLL), as required by θ ( k F ) in Eq.(4.1). Then, to understand the oscillations let us concentrate on the ρ H branch which is shown, together with M L (the in-medium mass), in Fig. 3 where it is clear that both quantities have an opposite oscillatory behavior. Theorigin of the oscillations in these quantities can be traced back to the fact that k max (the upper Landau level filled)decreases as the magnetic field increases. The first and second peaks, of the M L curve, correspond to the changefrom k max = 1 to k max = 0 for the up and down quark, respectively. For very low temperatures the value of µ atcoexistence decreases with B [20] so that, generally, k max and M must vary and when k max decreases, M increases.It then follows, from Eq. (4.1), that ρ B mu st decrease. When k max = 0 for both quark flavors there are no furtherchanges in the upper Landau level and the low temperature oscillations stop at eB > ∼ . m π .Let us now obtain the surface tension at vanishing temperature by first obtaining the difference between these twofree energies, ∆ f ( ρ ) ≡ f ( ρ ) − f M ( ρ ). Since f T ( ρ ) = ρµ ( ρ ) − P T ( ρ ) one can start by evaluating µ ( ρ ) and P ( ρ ) foruniform matter within the thermodynamically unstable region of the phase diagram. Figs. 4 and 5 show the resultsfor µ ( ρ ) and P ( ρ ) respectively and, as before, the continuous lines reflect the stable gap equation solutions and thedashed lines the unstable and metastable ones. It is then an easy task to obtain a (positive) deviation, ∆ f ( ρ ), whichdetermines the surface tension. Fig. 6 shows ∆ f ( ρ ) for B = 0, eB = 6 m π , and eB = 15 m π displaying the expectedoscillatory behavior around the B = 0 case. Fig. 7 shows the surface tension as a function of eB at T = 0 showingthat it oscillates around the B = 0 value for 0 < eB < ∼ m π before decreasing about 30% for 4 m π < ∼ eB < ∼ m π . Then,after reaching a minimum at eB ≈ m π it starts to increase continuously reaching the B = 0 value at eB ≈ m π .After that, only the LLL is filled and γ continues to grow with B .Finally, table I summarizes all our results for γ , when B = 0, eB = 6 m π , and eB = 15 m π , and also lists thecharacteristic values E g , and ρ g as well as the location of the critical point ( T c , µ c ), and the upper integral limit (seeEq. (2.3)), ρ H . For the present model approximation ρ L = 0 in all cases. The table also shows that the values of theconstituent quark mass, at T = 0 and µ = 0, grow with B in accordance with the magnetic catalysis phenomenon. æææææææææææææææææææææææææææææææææàààààààààààà ààààààààààààààààààààà Ρ BH M L eB (cid:144) m Π Ρ B H (cid:144) Ρ M L @ M e V D FIG. 3: The NJL model effective quark mass (squares) at the lowest value occurring at the transition, M L , and the highestcoexisting baryon density (dots), ρ HB (in units of ρ ), as functions of eB/m π at T = 0. The lines are shown just in order toguide the eye. Taken from Ref. [20]. B = Ρ B (cid:144) Ρ Μ @ M e V D eB = m Π Ρ B (cid:144) Ρ Μ @ M e V D eB = m Π Ρ B (cid:144) Ρ Μ @ M e V D FIG. 4: The chemical potential as a function of of ρ B /ρ for B = 0, eB = 6 m π , and eB = 15 m π . The continuous lines indicatethe gap equation stable solutions and the dashed lines the unstable and metastable ones. B. Thermal effects
Let us now investigate how thermal effects influence the interface tension since this quantity is expected to decreasewith increasing temperature because both the coexistence densities and the associated free energy densities movecloser together at higher T ; they ultimately coincide at T c where, therefore, the tension vanishes. This generalbehavior is confirmed by our calculations, as shown in Fig. 8. The temperature dependence of the surface tension maybe relevant for the thermal formation of quark droplets in cold hadronic matter found in “hot” protoneutron stars eB γ M T c µ c ρ HB /ρ ρ gB /ρ E g T = 0 for different values of eB (in units of m π ). The length parameter was takenas a = 0 .
33 fm. The characteristic energy density E g is given in MeV / fm , and the critical values µ c and T c are given in MeV.The effective magnetic quark masses M (at µ = 0) is also given in MeV while the resulting zero-temperature surface tension γ is given in MeV / fm . In all cases ρ LB = 0 and ρ = 0 . . æ æ B = - Ρ B (cid:144) Ρ P @ M e V (cid:144) f m D æ æ eB = m Π - Ρ B (cid:144) Ρ P @ M e V (cid:144) f m D æ æ eB = m Π - - Ρ B (cid:144) Ρ P @ M e V (cid:144) f m D FIG. 5: The pressure as a function of of ρ B /ρ for B = 0, eB = 6 m π , and eB = 15 m π . The continuous lines indicate thegap equation stable solutions and the dashed lines the unstable and metastable ones. The dotted lines joining the thick dotsrepresent the Maxwell construction. m Π m Π Ρ B (cid:144) Ρ D f @ M e V (cid:144) f m D FIG. 6: The quantity ∆ f as a function of ρ B /ρ for B = 0, eB = 6 m π , and eB = 15 m π . whose temperatures, T ∗ , are of the order 10–20 MeV [4, 28, 29]. For T ∗ the relevant value of γ T ∗ may be estimatedby using table I together with Fig. 8. The temperature dependence of the surface tension is also important in thecontext of heavy-ion collisions, because it determines the favored size of the clumping caused by the action of spinodalinstabilities as the expanding matter traverses the unstable phase-coexistence region [6]. C. Other possible effects
So far, our results for the surface tension were obtained within a certain model approximation, namely the standardtwo flavor NJL model at the mean field level. Therefore, one may wonder how other possibilities including a differentparametrization, strangeness, vector interactions, corrections beyond the MFA, and confinement, among others, wouldeventually influence our numerical predictions. Let us start this discussion with the parametrization issue in whichcase it becomes important to recall that, within the NJL model, a stronger coupling increases the first order transitionline in the T − µ plane. This fact is reflected by an increase of the coexistence region in the T − ρ B plane. Then, astronger coupling should produce a higher surface tension which is indeed the case, as demonstrated in Ref. [15] for0 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ eB (cid:144) m Π Γ @ M e V (cid:144) f m D FIG. 7: The surface tension at vanishing temperature, γ , as a function of eB (in units of m π ). The lines are present just inorder to guide the eye. æ æ æ æ æ æ æ æ æà à à à à à à à à à à à ì ì ì ì ì ì ì ì ì m Π m Π T @ MeV D Γ T @ M e V (cid:144) f m D FIG. 8: The surface tension, γ T , as a function of the temperature for for B = 0, eB = 6 m π , and eB = 15 m π . The lines arepresent just in order to guide the eye. B = 0. For example, taking Λ = 631 MeV, G Λ = 2 .
19, and m = 5 . T c = 46 MeVand µ c = 332 MeV while the effective quark mass value is M = 337 MeV (compare with our values in table I). Withthis parametrization, one obtains γ = 7 . / fm which is much smaller than our value, γ = 30 . / fm .On the other hand, the surface tension value is expected to increase by taking a higher coupling but one should alsoremember that the effective quark mass grows with G and, with the set adopted here, we already have M = 400 MeVwhich can be considered high enough . So, as far as the parametrization is concerned, our predictions could be lowered by adopting coupling values which predict smaller values for the effective quark mass.Next, let us point out that the presence of a repulsive vector channel may play an important role when treating the In most works the coupling is chosen so that M is about one third of the baryonic mass ( ≈
310 MeV). − G V ( ¯ ψγ µ ψ ) is usually added to the Lagrangiandensity describing the model [24, 30]. Then, regarding the phase diagram, it has been established that the net effect ofa repulsive vector contribution, parametrized by the coupling G V , is to add a term − G V ρ to the pressure weakeningthe first order transition [31]. In this case, the first order transition line shrinks, forcing the CP to appear at smallertemperatures, while the first order transition occurs at higher coexistence chemical potential values as G V increases.In this case, the coexistence region decreases (this situation will not be affected by the presence of a magnetic field[32]) and should produce an even smaller value for the surface tension.With respect to the MFA adopted here we believe that further improvements will only reduce the surface tensionsince evaluations performed with the nonperturbative Optimized Perturbation Theory (OPT), at G V = 0, have shown[26] that already at the first non trivial order the free energy receives contributions from two loop terms which are1 /N c suppressed. It turns out that these exchange (Fock) type of terms, which do not contribute at the large- N c (orMFA) level, produce a net effect similar to the one observed with the MFA at G V = 0. This is due to the fact thatthe OPT pressure displays a term of the form − G S / ( N f N c ) ρ where G S is the usual scalar coupling so that a vectorlike contribution can be generated by quantum corrections even when G V = 0 at the Lagrangian (tree) level. Therelation between the MFA (at G V = 0) and the OPT (at G V = 0) and their consequences for the first order phasetransition has been recently analyzed in g reat detail [33]. Based on this result one concludes that, in principle, theinclusion of corrections beyond the mean field level may contribute to further decrease the value of γ T .In stellar modeling, the structure of the star depends on the assumed EoS built with appropriate models whilethe true ground state of matter remains a source of speculation. It has been argued [34] that strange quark matter (SQM) is the true ground state of all matter and this hypothesis is known as the Bodmer-Witten conjecture. Hence,the interior of neutron stars should be composed predominantly of u, d, s quarks (plus leptons if one wants to ensurecharge neutrality). The question of how strangeness affects γ was originally addressed in Ref. [15] where the threeflavor NJL was considered yielding the value γ = 20 .
42 MeV / fm which is still within the lower end of estimatedvalues. Moreover, in their application to the three flavor Polyakov quark meson model, the authors of Ref. [16] haveconfirmed that the presence of strangeness should not affect the surface tension in a drastic way. Another importantissue, tread in Ref. [16], concerns confinement which has been considered by means of the Polyakov loop. Also, in thiscase the main outcome is that the surface tension value is not too much affected when the quark model is extendedby the Polyakov loop.Together, all these remarks indicate that our (low end) estimates for γ T are basically stable to the inclusion of morerefinements (such as strangeness and confinement) and can even be further lowered (e.g., by going beyond the meanfield level and/or by including a repulsive vector channel). V. CONCLUSIONS
In this work we have evaluated the surface tension related to the first-order chiral phase transition for two flavormagnetized quark matter by considering the NJL model in the MFA. To obtain this quantity we have used theprescription presented in Ref. [6] which is straightforward once the uniform-matter equation of state is available forthe unstable regions of the phase diagram. The surface tension determined in the present fashion is entirely consistentwith the employed model, including the approximations and parametrizations adopted. In practice one only needs toconsider all the solutions to the gap equation (stable, metastable and unstable) when generating the correspondingEoS. This method was previously employed to obtain the surface tension for the NJL in the absence of magnetic fieldsyielding γ < ∼ / fm which lies within the low end of available estimates ( γ ≈ −
300 MeV / fm ) and is inagreement with other recent predictions which employ effective quark models [14, 16]. The importance of this resultconcerns, for example, the possibility of a mixed phase occurring in hybrid stars since the existence of such a phaseis possible when the surface tension has a low value [18].Our results have shown that, when a magnetic field is present, the surface tension value presents a small oscillationaround the B = 0 value, for 0 < eB < ∼ m π . Then, it decreases for 4 m π < ∼ eB < ∼ m π reaching a minimum at eB ≈ m π where the value is about 30% smaller than the B = 0 result. After this point it starts to increasecontinuously reaching the B = 0 value at eB ≈ m π . This result allows to conclude that the existence of a mixedphase remains possible within this range of magnetic fields and can even be favored at the core of magnetars if B ∼ . × G (or, equivalently, eB ∼ m π ). At about twice this field intensity the surface tension starts toincrease rapidly with the magnetic field disfavoring the presence of a mixed phase within hybrid stars. The origin ofthis behavior can be traced back to the oscillations present in the coexistence region which is a quantity of centralimportance in the evaluation of γ T . We have also shown how the temperature affects this quantity by decreasing itsvalue towards zero which is achieved at T = T c , as e xpected. Other issues such as strangeness, the presence of arepulsive vector interaction, confinement, corrections to the MFA, as well as different parametrizations have also beendiscussed. We have argued that our surface tension values, which already rank at the low end of the available wide2range of predictions, will be little affected by strangeness and confinement and will be even lowered by the presenceof a repulsive vector term and/or by the inclusion of corrections beyond the mean field level so that a mixed phasewithin hybrid stars will be further favored by these improvements. On the other hand, with the adopted model, thesurface tension value could grow if one chooses a parametrization with a coupling greater than ours which in turnwould lead to very high effective quark masses. Acknowledgments
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