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Strange matter and strange stars in a thermodynamically self-consistent perturbation model with running coupling and running strange quark mass

A quark model with running coupling and running strange quark mass, which is thermodynamically self-consistent at both high and lower densities, is presented and applied to study properties of strange quark matter and structure of compact stars. An additional term to the thermodynamic potential density is determined by meeting the fundamental differential equation of thermodynamics. It plays an important role in comparatively lower density and ignorable at extremely high density, acting as a chemical-potential dependent bag constant. In this thermodynamically enhanced perturbative QCD model, strange quark matter still has the possibility of being absolutely stable, while the pure quark star has a sharp surface with a maximum mass as large as about 2 times the solar mass and a maximum radius of about 11 kilometers.


I. INTRODUCTION
In recent decades, strange quark matter (SQM) has been one of the most interesting and significant topics in nuclear physics [1]. Early in 1970's, the possible existence of a deconfined phase was proposed and studied [2][3][4][5]. Specially in 1984, it was speculated, based on elementary symmetry considerations, that SQM might be absolutely stable and thus have important consequences [6]. Soon after this it was shown in the MIT bag model that SQM is absolutely stable for a reasonable range of QCD-related parameters [7]. Since then a lot of works had been done on the properties and applications [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].
It is widely believed that the quantum chromodynamics (QCD) is the fundamental theory of strong interactions, and in principle, one could do detailed and comprehensive study on SQM by solving the motion equations of quarks and gluons. Unfortunately, however, QCD is, in fact, intractable in the nonperturbative regime presently. In particular at finite baryon chemical potential, there is a notorious sign problem where the Lattice Monte Carlo simulation is inaccessible [23]. Therefore, effective phenomenological models play crucial roles to extract and figure out the properties of strongly interacting matter. In the past years, a number of models have been applied with interesting results, such as the Nambu-Jona-Lasinio model [24,25], the global color symmetry model [26], the quasiparticle model [27][28][29][30][31][32][33], the mass-density-dependent model [34][35][36][37][38][39], the equivparticle model [40,41], the quark-cluster model [42], and so on.
Thermodynamic consistency is a fundamental requirement of phenomenological models [38]. In many important cases, an additional term to the thermodynamic potential density is necessary to maintain thermodynamic * Correspondence to: gxpeng@ucas.ac.cn consistency. In an important version of the quasiparticle model, for example, the additional term is needed in both the zero [43] and finite temperature [44] cases. In the equivparticle model with confinement by the density dependence of quark masses, an additional term also appears in the thermodynamic potential density to have full thermodynamic consistency [40,41].
Because of asymptotic freedom, the perturbative calculation of QCD is reliable at very high densities. The thermodynamic potential density of cold quark matter was calculated for massless quarks in Refs. [45][46][47]. These results were applied to study quark stars to the first order in QCD coupling in Refs. [48,49], to the second order in [50], with finite-mass effect of strange quarks considered in Refs. [51][52][53].
The validity of a perturbative theory requires a small coupling with which the perturbative series is obtained. Different from quantum electrodynamics (QED), however, the QCD coupling is running, i.e., it is not that small when the density is not extremely high. In this case, one will meet thermodynamic problem when one naively extends the applicable range of density [54].
One way to solve this problem is to add an additional term to the thermodynamic potential density, similar to that of the popular quasiparticle model [55][56][57][58] and the equivparticle model [40,41]. This way of extending the applicable range of the perburbative calculation was shown to be reasonable for the one-flavor case [54]. It has also been shown, for the massless two-flavor case, that the renormalization subtraction point should be taken as a function of the summation of the biquadratic chemical potentials while the additional term not only keeps the thermodynamics self-consistent, but also produces reasonable results [59]. In the present paper, we extend this thermodynamically enhanced perturbative QCD (EPQ) model to the actual SQM with massless up (u) and down (d) quarks plus massive strange (s) quarks. It is found that the additional term takes an important role at lower densities, acting as a chemical-potential dependent bag constant. The equation of state (EOS) of SQM becomes stiffer, and accordingly the maximum mass of strange stars is as large as two times the solar mass.
The paper is organized as follows. In the next section II, we give the conventional perturbative treatment and demonstrate the thermodynamic inconsistency in its naive extension to lower densities. Then in Sec. III, we determine a chemical-potential dependent bag-like coupling constant which makes the thermodynamic treatment self-consistent. After that we study, respectively, the properties of SQM and the structure of compact stars with the new EPQ model in Sec. IV and V. Finally the section VI is a short summary.

II. THE CONVENTIONAL PERTURBATION MODEL AND INCONSISTENCY OF ITS NAIVE EXTRAPOLATION
Let's start our work from the perturbative expansion of the thermodynamic potential density of cold quark matter with two-flavor massless light quarks plus one massive strange quark. According to Eqs. (1) and (2) in Refs. [51,52], we have the perturbative contribution to the thermodynamic potential density as where, to the first order, the contributions from massless up and down quarks are respectively (2) and that from the massive strange quarks is [51,53] Here µ u , µ d , and µ s are the chemical potentials of up, down, and strange quarks respectively, u is the renormalization subtraction point, α ≡ α s /π = g 2 /(4π 2 ) is the running coupling, and ach x ≡ ln(x + √ x 2 − 1) is the inverse hyperbolic cosine function. Because the mass of u and d quarks is much smaller than that of s quarks, we consider only the mass effect of strange quarks. For simplicity, we have used the notation ν s ≡ µ 2 s − m 2 s , which can be regarded as the fermion momentum of s quarks. Because the electron does not participate in the strong interactions, its contribution to the thermodynamic potentail is then The number densities of u and d quarks and electrons are, respectively while that of s quarks is Here we should keep in mind that all terms with order in the coupling higher than unity have been discarded because we assume the perburbative expression is merely valid to leading order.
The thermodynamic potential density of the whole system composed of u, d, s quarks and electrons, given by the sum of Eq. (2) to Eq. (4), depends on the quark chemical potentials µ u , µ d , µ s , µ e , and u both explicitly and implicitly through the scale dependence of α(u) and the mass m s (u).
The running coupling α(u) and the running mass m s (u) of strange quarks are determined by the following renormalization group (RG) equations where N is the loop number, while the beta and gamma functions, β(α) and γ(α), are presently known to fourloop level, given by the corresponding beta and gamma coefficients, i.e., β i and γ i . The original ones were given in the minimum subtraction scheme or its modified version (MS). The coupling and masses given with these β i and γ i are, in principle, not continuous at heavy quark thresholds. In order to give a continuous coupling and a continuous strange quark mass, the beta and gamma coefficients should be recombined. For the number of colors N c = 3, these coefficients are provided in the appendix A. Comparing these coefficients with the original beta and gamma functions [60,61], one finds that β 0 , β 1 , γ 0 , and γ 1 are not changed and thus universal [62], while modifications are necessary for β j≥2 and γ j≥2 . The exact solutions of Eqs. (7) and (8) can be obtained by separation of variables, as shown in the appendix A. At one-loop level, the running coupling and running quark mass of strange quarks are, respectively, given by , m s =m s α γ0/β0 , where β 0 = 11/4 − N f /6, γ 0 = 1, Λ andm s are the QCD scale parameters respectively for the coupling and strange quark mass. With the requirement of continuity at the threshold of heavy quark masses and the initial condition α s (M Z ) = 0.1185 (where M Z =91.1876 MeV is the mass of Z bosons) and m s (2GeV) = 93.5 MeV [63], one can get distinct coupling scale Λ N f and mass scalem s,N f for different effective number of flavors, i.e. Λ 3−6 andm s,3−6 , respectively corresponding to u in different range, i.e., u < m c , GeV, m b = 4.18 GeV, m t = 173.21 GeV [63]. The results to four-loop level and for different number of flavors are given in TABLE I. Because we are considering three flavors to the leading order, in the following calculations we take Λ = Λ 3 = 146 MeV andm s =m s,3 = 280 MeV, respectively. Now we need to choose the relation between the renamalization point u and the chemical potentials. In principle, the choice is not unique. In Ref. [51], it is chosen to be To maintain weak equilibrium, the chemical potentials satisfy Furthermore, SQM should be in the state of charge neutrality, i.e., Another condition for the quark system is the baryon number conservation, reading For a given baryon number density n b we can solve Eqs. (11)- (13) to obtain all the relevant chemical potentials. Then the pressure P and the energy density E are given by where i goes over all flavors and electron.
To check thermodynamic consistency of phenomenological models, a discriminant ∆, as a function of the baryon number density, was introduced in Ref. [41] as where E and P are respectively model-given energy density and pressure. For any thermodynamically consistent models, the discriminant vanishes at arbitrary density. In order to show the inconsistency degree of the above described pure perturbation model, we show in FIG. 1 the density behavior of the ratio ∆/E as a function of the density n b . From this figure, one can easily find that the discriminant ratio decreases monotonically with increasing density. This is a clear demonstration that the thermodynamics at higher density nearly consistent while the thermodynamic inconsistency becomes more and more serious with decreasing density.
One might think that this problem can be solved by adding a pure constant B 0 to the energy density (subtracting from the pressure) and interpreting it as the vacuum energy density, just like that had been done in the original bag model. In FIG. 2, we plot the energy per baryon (left axis) and pressure (right axis) as function of the density for 4 √ B 0 = 135 MeV. It is obvious in this case that there exists a minimum energy per baryon (marked with a solid triagle) and zero pressure (the small open circle). As emphasized in Ref. [38], also directly seen from the first equality of Eq. (16), these two points should appear exactly at the same density. However, FIG. 2 clearly shows that they are obviously deviate from each, contradicting the fundamental thermodynamics. This is understandable from the second equality of Eq. (16): adding a constant to E and subtracting it simultaneously from the pressure P do not influence the value of the discriminant ∆. In fact, if one draws the density behavior of ∆, one will find that it is totally overlapped with the case without the constant.
In the following section we will find an additional term that depends on chemical potentials. The chemicalpotential-dependent term can be neglected at high density but it takes an important role at lower density, solving the thermodynamic inconsistency very nicely. The key point is that the renormalization subtraction u as a function of the chemical potentials can not be arbitrarily taken, such as that in Eq. (10). Instead, we choose it to satisfy an equation obtained by the requirement of thermodynamic consistency.

III. EXTRAPOLATION WITH THERMODYNAMIC CONSISTENCY
The thermodynamic potential density of a cold quark system from perturbative QCD to order N can be generally written as where α = α s /π = g 2 /(4π 2 ) is the QCD running coupling, m s is the running mass of a strange quark. They satisfy the renormalization group equation given in Eq. (7) and (8).
The corresponding number density for quark flavor q=u, d, s can be easily obtained by the normal thermodynamic relation n q = −dΩ pt /dµ q . Also at the order N , it gives We write the whole thermodynamic potential density of the system as Here Ω pt N is the perturbative contribution, while Ω ′ is the non-perturbative contribution. To determine Ω ′ , we require that it makes Ω satisfy the fundamental thermodynamic equation where S is the entropy density at temperature T . At zero temperature the first term vanishes and we have dΩ = − i n i dµ i . Substituting Eqs. (20), (17), and (18) into this equality, we immediately obtain where the partial derivatives are The double summation in Eqs. (23)-(25) is given by Therefore, the additional term Ω ′ is given by a path integral as where µ 0 = (µ u0 , µ d0 , µ s0 ) is a starting point for the path integral which is fixed to be µ u0 = µ d0 = µ s0 = 313 MeV in the present calculation, while its moving effect is boiled down to another constant B 0 . As everyone knows, the thermodynamic potential is a state function. Therefore, Ω ′ should be independent of the path, namely, the integration in Eq. (27) is pathindependent. This requires that the integrands satisfy the Cauchy conditions It is easy to prove that only two of them are independent.
In the present paper, let's take the first-order of Ω pt N as an example , i.e. Ω pt 1 , with running coupling α and running mass m s expanded to first order which are explicitly given by Eq. (9). In this case, we have With the notation τ ≡ 4 µ 4 u + µ 4 d + µ 4 s and the expressions of ω 0 and ω 1 , i.e.
one can derive the explicit expression of f 1,0 , giving Substituting Eq. (29) into the first two Cauchy conditions, we have From Eq. (31), one can get the partial derivatives of f 1,0 with respect to µ u and µ d respectively, i.e.
Using Eq. (33), the first equation in Eq. (32) becomes This equation means that the solution of u is a function of µ s and ρ ≡ 4 µ 4 u + µ 4 d , i.e. u = u(ρ, µ s ). However, this function is not necessarily explicit, and can be generally implicit. So we assume it is determined by the following implicit equation In order to find the form of Φ, we give the partial derivatives of u with respect to µ d and µ s , i.e.
where the notations Φ x ≡ ∂Φ/∂x (x = ρ, u, µ s ) have been used. Then substituting Eq. (36) to the second equation in Eq. (32) leads to where ∂f1,0 ∂µs can be obtained from Eq. (31), i.e., For Eq. (37) to be fulfilled, we choose Solving the equalities in Eq. (39), we can find the solution where φ(u) is an arbitrary function of u. In fact, integrating the first equality in Eq. (39), one can get the solution as Φ(ρ, µ s , u) = ρ 4 + ϕ(µ s , u), where the integration constant ϕ (with respect to the variables µ s and u) can be obtained by substituting into the second equality of Eq. (39) as which gives the solution in Eq. (40) immediately. For convenience and simplicity, we have chosen the function φ(u) = N f (u/C) 4 with constant C being a model parameter which can be fixed in a reasonable region by the common knowledge of modern nuclear physics. So, from the solution of Eq. (32), one can find that to solve the problem of inconsistency in thermodynamics, the relation between the renormalization subtraction point u and chemical potentials should be obtained by solving the following equation As is well known, the mass effect from strange quarks can be ignored at very high density. In this case the relation between the renormalization subtraction point and chemical potentials can be given in a simple form, i.e., which can be obtained by taking m s → 0 in Eq. (41). In order to get the EOS, in general one should numerically solve Eq. (41). Then substituting Eq. (29) to Eq. (27) and numerically integrating it, one can get the quantity Ω ′ which is essential in fixing the problem of thermodynamic inconsistency. And the partial derivatives of renormalization subtraction point u with respect to chemical potentials µ q (q = u, d, s) in Eq. (29) are where ∂f 1,0 /∂µ q are already given in Eqs. (33) and (38).
Other relevant derivatives of f 1,0 in Eq. (43) are, respectively The remaining derivatives on the right-hand side of Eq.

IV. EOS OF STRANGE QUARK MATTER
As usually done, we assume SQM to be a mixture of interacting quarks and free electrons. So the total thermodynamic potential density Ω tot reads where the first term is the perturbative contribution in Eq. (17) whose concrete form is given to leading order in Eqs. (2) and (3). The second term is the contribution from electrons treated as free particles because they do not participate in the strong interactions, and the last term is given in Eq. (27), determined by thermodynamic consistency requirement to consider non-perturbative effect. For a given baryon number density, one can solve Eqs. (11)-(13) with the help of Eqs. (5) and (6) to obtain the relevant particle chemical potentials, then all other quantities can be thermodynamically obtained. But in the present model, there are still two parameters, B 0 and C, to be determined by stability arguments. As is well known, the energy per baryon of two-flavor quark matter should be bigger than 930 MeV (E/n b > 930 MeV), in order not to contradict standard nuclear physics. Therefore, the shaded region in FIG. 3 is forbidden. If the energy per baryon of three flavor quark matter is less than 930 MeV, the SQM is absolutely stable; if it is bigger than 930 MeV, but smaller than 939 MeV, the SQM is metastable; otherwise, the SQM is untable. These different regions are indicated in FIG. 3.
To investigate the properties of SQM in the present EPQ model, we choose three typical sets of parameters in the absolutely stable region, i.e., (  the value of ∆/E for the present model is zero at all density. In addition, the three minimum points in this figure correspond to absolutely stable SQM, as expected. It is also found that the minimum energy per baryon in fact becomes bigger with decreasing C and/or increasing B 0 . FIG. 5 gives the EOS of SQM. The stiffness of EOS obviously varies with the model parameters C and B 0 , e.g., EOS would become stiffer with bigger C and/or smaller B 0 . In the next section we will see that this means a bigger maximum mass of compact stars (see FIG. 8).
To check the impact of the additional term Ω ′ , we plot, in FIG. 6, the density behavior of the velocity of sound calculated by We have noted that the density behavior of sound velocity is greatly affected by the stiffness of EOS, and it is understandable that stiffer EOS corresponds to fast velocity of sound. At very high density, however, they all approach to the ultra relativistic case. In addition, we would like to point out that the sound velocity is independent of the parameter B 0 .
FIG . 7 shows the density behavior of Ω ′ and its relative importance. From this figure one can see that although Ω ′ increases with increasing density, the relative importance decreases with increasing density. That means Ω ′ plays a relatively important role at lower density while it is ignorable at high density. In this regard it is similar to a chemical-potential-dependent bag constant.
In general case, such a compact object may be a hybrid star with pure quark core and hadronic crust [8]. Because SQM can be self-bound, i.e., its internal pressure can be zero at a definite density (see the minima in Fig. 4), the whole star can be converted to a pure quark star, for example, as a strong deflagration process during a few milliseconds [74], or seeded with slets [75] by the self-annihilating weakly interacting massive particles [76], etc.
In the preceding sections, we have developed an enhanced version of a perturbative QCD treatment of the dense quark matter by thermodynamic consistency requirement. Now we apply it to study the structure of quark stars. For this purpose we should solve the Tolman-Oppenheimer-Volkov (TOV) equation [77] dP dr = − GmE r 2 with the subsidiary condition where G = 6.707 × 10 −45 MeV −2 is the gravitational constant, r is the distance from the center of a quark star, and P and E are the pressure and energy density with their mutual relation given by EOS. One can refer to Ref. [38] for a concise process of how to solve this equation.
On application of the equations of state in Fig. 5, we can get the mass-radius relation in FIG. 8 for   parameter sets indicated in the legend. This figure shows several feathers of quark stars.
(1) The radius of a quark star can be in principle very small, i.e., there is no lower bound to the radius. This is very different from the normal neutron stars whose radius are normally greater than a critical value.
(2) For a given set of parameters, the star radius first increases with increasing the star mass, until a maximum radius is reached. After that, the radius decreases until the maximum star mass is arrived.
(3) The maximum star mass depends on parameters. It actually increases with increasing C, while decreases with increasing B 0 . Specially for the typical parameters C = 2 and B 0 = (135 MeV) 4 , The maximum mass is about two times the solar mass, consistent with the recent high-mass observations [69,70].
To understand the existence of a maximum star mass, we show, in FIG. 9 for the parameter set A, the star mass as a function of the central density, with the central pressure also given on the right axis. One can easily find that the star mass first increases with increasing the central density to a critical density n max . After this density, the star mass decreases with increasing the central density, and the star itself becomes mechanically unstable. The central pressure is always an increasing function of the central density. It approaches to zero if the quark star mass becomes zero. Please note, the corresponding central density to zero pressure is nonzero. Instead, it is a definite value corresponding to the surface density of the quark star. The density of a quark star is not uniformly distributed. In FIG. 10, we plot the density profiles with different panels for different parameters. For each parameter set, the upmost curve corresponds to the star with the maximum mass, r = 0 corresponds to the central density. At the star surface, the pressure is zero. This point corresponds to the minimum energy per baryon in FIG. 4. Therefore, the surface density of the quark star is not zero, i.e., the star has a sharp surface. In Tab. II, we list some characteristic quantities of quark stars for each parameter set, including the maximum star mass to the solar mass M max /M ⊙ , the corresponding radius R(M max ), the highest central density n max , the surface density n 0 , and the minimum energy per baryon E 0 /n 0 .
It should be noted that the observation of quark stars having a sharp surface depends very much on the model assumption. Especially when one doesn't have a colorflavor locked phase throughout, the star might have a crust of ordinary matter supported by electrons extending beyond the quark surface. Or the outer layers of the quark star might fragment to strangelets, somewhat similar to a normal neutron star crust. Also, for some parameters, e.g. the case C, the surface density is closer to the normal nuclear saturation density, which might be an indication of phase transition to nuclear matter. To understand possible phase transition in a mixed star, it is necessary to investigate phase equilibrium condition in the phase boundary [8,40].

VI. SUMMARY
The perturbative QCD is important to study strongly interacting matter. However, its naive extension to comparatively lower density has serious thermodynamic inconsistency problems due to the running of the QCD coupling and/or quark masses. We have tried to extend it by including an additional term in the thermodynamic potential density. The additional term is determined by the fundamental differential equation of thermodynamics. It takes an important role at comparatively lower density, but ignorable at high density, playing the role of a chemical-potential-dependent bag constant.
On application of the thermodynamically EPQ model, we study the properties of SQM and structure of quark stars. It is found that SQM still has the possibility of being absolutely stable in the present model, i.e. the minimum energy per baryon could be less than 930 MeV with zero internal pressure. This leads to the maximum mass of quark stars as large as two times the solar mass. The quark stars in the present model has several features, such as a sharp surface, no lower bound for the radius, the maximum mass of about two times the solar mass and a maximum radius of about 11 kilometers, etc.
Naturally, there several aspects not included in the present investigations, e.g. the color superconductivity [32,78,79], a strong external magnetic field [58,[80][81][82], etc. Furthermore, only zero temperature has been considered in the present paper. To investigate the properties of hot quark matter, such as the quark gluon plasma produced in high energy heavy ion collisions [83,84], the inclusion of finite temperature is of crucial importance. Therefore, more careful studies are necessary in the future. pointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and the Shu Guang project of Shanghai Municipal Education Commission and Shanghai Education Development Foundation.

Appendix A: Matching-invariant beta and gamma functions and solutions of the renormalization equations
A matching-invariant beta function was previously derived in Ref. [85]. The beta and gamma functions which give matching-invariant running coupling and running quark masses in QCD can be similarly obtained. In the present context, the beta and gamma coefficients are given by In order to have full thermodynamic consistency in the present model, we need the exact solutions of the renormalization equation (7) at a given loop level. For this purpose, we can apply the approach of variable separation which gives where the acute beta function is defined to beβ(α) ≡ β 0 /β(α) − β 1 /(β 0 α) + 1/α 2 . Or, using the expression of the beta function, one haś Integrating both sides of Eq. (A4) gives ln where the function W N (α) is given by W N (α) = β 0 α 0β (x) dx. For example, it is not difficult to give W 2 (α) = −β 1 ln 1 + β 1 β 0 α , (A7) The Λ in Eq. (A6) is a QCD scale for the running coupling. In principle, it is an integration constant. The choice in Eq. (A6) is to be consistent with the conventional series expansion. For a chosen u, the corresponding α is obtained by numerically solving the algebraic equation (A6). At the one-loop level, the function W N (α) becomes W 1 (α) = −β 1 ln(β 0 α). In this case, an explicit solution can be easily obtained, as given in the first equality of Eq. (9).