Quark mass scaling and properties of light-quark matter

Quark mass scaling and properties of light-quark matter 1 first-author 1 2 3 corresp Corresponding author gxpeng@ucas.ac.cn Corresponding author gxpeng@ucas.ac.cn gxpeng@ucas.ac.cn 1 1 4 5 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Theoretical Physics Center for Science Facilities, Institute of High Energy Physics, Beijing 100049, China Theoretical Physics Center for Science Facilities, Institute of High Energy Physics, Beijing 100049, China Synergetic Innovation Center for Quantum Effects and Application, Hunan Normal University, Changsha 410081, China Synergetic Innovation Center for Quantum Effects and Application, Hunan Normal University, Changsha 410081, China Department of Physics and Astronomy, University of Catania, Via S. Sofia 64, I-95125 Catania, Italy Department of Physics and Astronomy, University of Catania, Via S. Sofia 64, I-95125 Catania, Italy Laboratori Nazionali del Sud, INFN-LNS, Via S. Sofia 62, I-95123 Catania, Italy Laboratori Nazionali del Sud, INFN-LNS, Via S. Sofia 62, I-95123 Catania, Italy We study the properties of two-flavor quark matter in the equivalent particle model. A new quark mass scaling at finite temperature is proposed and applied to the thermodynamics of two-flavor quark matter. It is found that the perturbative interaction has strong effect on quark matter properties at finite temperature and high density. The pressure at the minimum free energy per baryon is exactly zero. With increasing temperature, the energy per baryon increases while the free energy per baryon decreases. Quark matter Equation of state Quark mass scaling 1 Introduction 1 In many fields of nuclear physics, e.g. the Quantum Chromodynamics (QCD) phase diagram and structure of compact stars, one needs the equation of state of quark matter [ 1 1 ]. In principle, we have the fundamental theory of strong interactions, i.e. QCD. Due to the non-perturbative QCD interaction at relatively lower densities on one hand, and the so-called "sign problem" on the other hand, QCD can not be studied from first principles at finite baryon density. Therefore, one usually resort to various phenomenological models, e.g., Nambu and Jona-Lasinio model [ 2 2 ], perturbation model [ 3 3 - 5 5 3-5 ], field correlator method [ 6 6 ], quark-cluster model [ 7 7 ], and many other models [ 8 8 - 18 18 8-18 ]. A useful way to include interactions among quarks is to use medium-dependent quark masses. In one class of models, namely the quasiparticle models, the quark mass depends on chemical potential and/or temperature [ 19 19 - 27 27 19-27 ]. In another case, the quark mass depends on density and/or temperature [ 28 28 - 31 31 28-31 ]. These kinds of models were previously called the density-dependent (density- and temperature-dependent) quark mass models, but the thermodynamic treatment is inconsistent. In the case of chemical-potential dependent masses, one can have thermodynamic consistency by adding an additional term to the thermodynamic potential density [ 20 20 ]. In the density-dependent case, it is now clear that the original chemical potentials should be replaced with effective ones when the quark masses become density- and temperature- dependent [ 32 32 ]. The most recent version is fully thermodynamically self-consistent, and is called the equivparticle model [ 33 33 ]. Another important issue in the equivparticle model is the quark mass scaling, i.e. how to parameterize the density dependence of the quark masses. Initially, people mainly emphasize quark confinement, and the interaction part in the quark mass is parameterized to be inversely proportional to the baryon number density [ 28 28 ] for light quarks, and later extended to strange quarks [ 29 29 ]. According to the in-medium chiral condensate, it was shown that the interaction quark mass should be inversely proportional to the cubic root of the density [ 14 14 ]. It is true that confinement interactions dominates at lower densities. With increasing densities, however, the perturbative interaction sets in, and becomes more and more important, and thus should be included. Recently, a new quark-mass scaling, considering both the confinement and perturbative interactions, was derived, and applied to the investigation of SQM and strange stars [ 33 33 ]. But the important temperature effect was not considered there. The purpose of the present paper is to study the properties of two-flavor quark matter with the equivparticle model with the inclusion of both confinement and first-order perturbative interaction at finite temperature and density. In Sect. 2, we will discuss the quark mass scaling, while in Sect. 3, we give the necessary thermodynamic framework and numerical results. Finally we summarize the paper in Sect. 4. 2 Quark mass scaling 1 Originally, the light quark mass m q ( q =u/d) was assumed to be inversely proportional to density, i.e. [ 28 28 ] <math display="block" id="M1"><mrow><msub><mi>m</mi><mrow><mtext>u/d</mtext></mrow></msub><mo>=</mo><mfrac><mi>B</mi><mrow><mn>3</mn><msub><mi>n</mi><mtext>b</mtext></msub></mrow></mfrac><mn>,</mn></mrow></math> where B is the bag constant, and n b is the baryon number density of quark matter. It was soon extended to including strange quarks [ 30 30 ] <math display="block" id="M2"><mrow><msub><mi>m</mi><mtext>s</mtext></msub><mo>=</mo><msub><mi>m</mi><mrow><mi>s</mi><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>B</mi><mrow><mn>3</mn><msub><mi>n</mi><mtext>b</mtext></msub></mrow></mfrac><mn>.</mn></mrow></math> Based on the in-medium chiral condensates and linear confinement, a new scaling was derived [ 14 14 ] <math display="block" id="M3"><mrow><msub><mi>m</mi><mi>q</mi></msub><mo>=</mo><msub><mi>m</mi><mrow><mi>q</mi><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>D</mi><mrow><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mn>.</mn></mrow></math> This cubic scaling has been extensively applied to the investigations of SQM-related physics, e.g. the QCD phase diagram [ 32 32 ], properties of SQM and slets [ 34 34 ] at zero and finite temperature [ 16 16 , 35 35 ], the damping time scale due to the coupling of the viscosity and r mode [ 36 36 ], and the quark-diquark equation of state and compact star structure [ 37 37 ] etc. The energy per baryon of quark matter becomes infinite for small density, describing effectively quark confinement. Although the confinement interaction is dominant at lower density, perturbative interactions become more and more important with increasing density. Recently, a new quark mass scaling considers both confinement and first-order perturbative interactions has been derived [ 33 33 ] as <math display="block" id="M4"><mrow><msub><mi>m</mi><mi>q</mi></msub><mo>=</mo><msub><mi>m</mi><mrow><mi>q</mi><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>D</mi><mrow><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mo>+</mo><mi>C</mi><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mn>.</mn></mrow></math> The cubic-root scaling in Eq. (3) was extended to include finite temperature [ 16 16 ] as <math display="block" id="M5"><mrow><msub><mi>m</mi><mi>q</mi></msub><mo>=</mo><msub><mi>m</mi><mrow><mi>q</mi><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>D</mi><mrow><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mrow><mn>8</mn><mi>T</mi></mrow><mrow><mi>λ</mi><msub><mi>T</mi><mi>c</mi></msub></mrow></mfrac><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mi>λ</mi><mfrac><mrow><msub><mi>T</mi><mi>c</mi></msub></mrow><mi>T</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> where λ 1.6, and T c is the critical temperature from where the confinement effect becomes ignorable. The advantage of the extension to finite temperature in Eq. (5) is that it is in consistent with the string theory at lower temperature. An obvious disadvantage is that when one extends it to high temperature, it approaches to zero at T = T c , which was previously regarded as a critical temperature. When T > T c , the expression becomes negative, and one has to take a zero value for the interaction part. We can now write Eq. (5) in another form as <math display="block" id="M6"><mrow><msub><mi>m</mi><mi>q</mi></msub><mo>=</mo><msub><mi>m</mi><mrow><mi>q</mi><mn>0</mn></mrow></msub><mo>+</mo><mfrac><mi>D</mi><mrow><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><msup><mrow><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>8</mn><mi>T</mi></mrow><mrow><mi>λ</mi><msub><mi>T</mi><mi>c</mi></msub></mrow></mfrac><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mi>λ</mi><mfrac><mrow><msub><mi>T</mi><mi>c</mi></msub></mrow><mi>T</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mo>]</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mn>.</mn></mrow></math> The new expression in Eq. (6) equals to that in Eq. (5) at lower temperature. At higher temperature, however, they act very differently: the former becomes inversely proportional to temperature while the latter is zero. Observing Eq. (4), we notice that, except for an adjustable constant, the confinement and perturbative interaction terms are reciprocal. Therefore, inspired by Eqs. (4) and (6) we assume that at finite temperature and density quark mass behaves: <math display="block" id="M7"><mtable columnalign="left"><mtr><mtd><msub><mi>m</mi><mi>q</mi></msub><mo>=</mo><msub><mi>m</mi><mrow><mi>q</mi><mn>0</mn><mo>+</mo></mrow></msub><mfrac><mi>D</mi><mrow><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>8</mn><mi>T</mi></mrow><mi>Λ</mi></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>Λ</mi><mo>/</mo><mi>T</mi></mrow></msup></mrow><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr><mtr><mtd><mo>+</mo><mi>C</mi><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>8</mn><mi>T</mi></mrow><mi>Λ</mi></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>Λ</mi><mo>/</mo><mi>T</mi></mrow></msup></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> where Λ is a temperature scale parameter, comparable with Λ QCD . Comparing the low density limit of Eq. (7) with Eq. (6), we should have Λ= λT c . We thus take Λ =280 MeV in the present calculations. The confinement parameter D and the perturbative strength C should be chosen to meet the condition that the minimum energy per baryon of light-quark matter is greater than 930 MeV at zero temperature, in order not to contradict with the conventional nuclear physics. According to our recent study, D 1/2 should be bigger than 120 MeV [ 33 33 ] but smaller than 270 MeV [ 16 16 ], and C is smaller than 1 [ 33 33 ]. For the present case of the two-flavor quark matter, we take the modest values D 1/2 =160 MeV and C =0.6. This choice has the property that the confinement term decreases up to a flex temperature point at about 155 MeV. In Fig. 1 Fig. 1 , we show the quark mass as a function of the temperature at the density n b =0.5 fm -3 for the parameters D 1/2 =160 MeV and C =0.6 with a solid curve. The confinement and perturbative contributions are also plotted, respectively, with dotted and dashed curves. We notice that with increasing temperature, the confinement term in the quark mass becomes less important than the perturbative one. For the gluon mass, we, according to the first-order perturbative result, use <math display="block" id="M8"><mrow><msup><mrow><mo stretchy="false">(</mo><msub><mi>m</mi><mtext>g</mtext></msub><mo>/</mo><mi>T</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><mi>η</mi><mi>α</mi><mtext> </mtext><mi>θ</mi><mo stretchy="false">(</mo><mi>T</mi><mo>−</mo><msub><mi>T</mi><mi>c</mi></msub><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where T c is the critical temperature of a pure SU(3) gluon gas. The step function, θ , is unity with a non-negative argument and zero otherwise. The parameter, η , is taken to be η =15 in the present calculations, while for the running coupling, we use a fast convergent expansion [ 42 42 ]: <math display="block" id="M9"><mrow><mi>α</mi><mo>=</mo><mfrac><mrow><msub><mi>β</mi><mn>0</mn></msub></mrow><mrow><msubsup><mi>β</mi><mn>0</mn><mn>2</mn></msubsup><mi>ln</mi><mo stretchy="false">(</mo><mi>u</mi><mo>/</mo><msub><mi>Λ</mi><mi>T</mi></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>β</mi><mn>1</mn></msub><mi>ln</mi><mi>ln</mi><mo stretchy="false">(</mo><mi>u</mi><mo>/</mo><msub><mi>Λ</mi><mi>T</mi></msub><mo stretchy="false">)</mo></mrow></mfrac><mn>,</mn></mrow></math> where the beta coefficients are taken to be β 0 =11/2- N f /3 and β 1 =51/4-(19/12) N f . In order to include nonperturbative effects, the renormalization subtraction point is assumed to vary linearly with the temperature as <math display="block" id="M10"><mrow><mi>u</mi><mo>/</mo><msub><mi>Λ</mi><mi>T</mi></msub><mo>=</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><mi>x</mi></mrow></math> with c 0 =1, c 1 =1/2, and x T / T c . To calculate the properties of quark matter in the equivparticle model, we need to use the density and temperature derivatives of the quark mass, which can be easily obtained from Eq. (7), giving <math display="block" id="M11"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mo>∂</mo><msub><mi>m</mi><mi>q</mi></msub></mrow><mrow><mo>∂</mo><msub><mi>n</mi><mtext>b</mtext></msub></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mi>D</mi><mrow><mn>3</mn><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>8</mn><mi>T</mi></mrow><mi>Λ</mi></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>Λ</mi><mo>/</mo><mi>T</mi></mrow></msup></mrow><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mtd></mtr><mtr><mtd><mo>+</mo><mfrac><mi>C</mi><mrow><mn>3</mn><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>2</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>8</mn><mi>T</mi></mrow><mi>Λ</mi></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>Λ</mi><mo>/</mo><mi>T</mi></mrow></msup></mrow><mo>)</mo></mrow><mn>,</mn></mtd></mtr></mtable></math> and <math display="block" id="M12"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mo>∂</mo><msub><mi>m</mi><mi>q</mi></msub></mrow><mrow><mo>∂</mo><mi>T</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><mn>8</mn><mi>D</mi></mrow><mrow><mi>Λ</mi><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup></mrow></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mi>Λ</mi><mi>T</mi></mfrac></mrow><mo>)</mo></mrow><mfrac><mrow><msup><mi>e</mi><mrow><mo>−</mo><mi>Λ</mi><mo>/</mo><mi>T</mi></mrow></msup></mrow><mrow><msup><mrow><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mrow><mn>8</mn><mi>T</mi></mrow><mi>Λ</mi></mfrac><msup><mi>e</mi><mrow><mo>−</mo><mi>Λ</mi><mo>/</mo><mi>T</mi></mrow></msup></mrow><mo>)</mo></mrow></mrow><mn>2</mn></msup></mrow></mfrac></mtd></mtr><mtr><mtd><mo>+</mo><mn>8</mn><mi>C</mi><msubsup><mi>n</mi><mtext>b</mtext><mrow><mn>1</mn><mo>/</mo><mn>3</mn></mrow></msubsup><mrow><mo>(</mo><mrow><mfrac><mn>1</mn><mi>T</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><mi>Λ</mi></mfrac></mrow><mo>)</mo></mrow><mi>exp</mi><mrow><mo>(</mo><mrow><mo>−</mo><mfrac><mi>Λ</mi><mi>T</mi></mfrac></mrow><mo>)</mo></mrow><mn>.</mn></mtd></mtr></mtable></math> The temperature derivative of the gluon mass can be similarly obtained from Eq. (8), i.e., <math display="block" id="M13"><mrow><mfrac><mrow><mtext>d</mtext><msub><mi>m</mi><mtext>g</mtext></msub></mrow><mrow><mtext>d</mtext><mi>T</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>m</mi><mtext>g</mtext></msub></mrow><mi>T</mi></mfrac><mrow><mo>(</mo><mrow><mn>1</mn><mo>+</mo><mfrac><mi>x</mi><mn>2</mn></mfrac><mfrac><mrow><mtext>d</mtext><mi>ln</mi><mi>α</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where <math display="block" id="M14"><mrow><mfrac><mrow><mtext>d</mtext><mi>ln</mi><mi>α</mi></mrow><mrow><mtext>d</mtext><mi>x</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mrow><msub><mi>c</mi><mn>1</mn></msub><mi>α</mi></mrow><mrow><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><mrow><msub><mi>β</mi><mn>0</mn></msub><mo>+</mo><mfrac><mrow><msub><mi>β</mi><mn>1</mn></msub></mrow><mrow><msub><mi>β</mi><mn>0</mn></msub></mrow></mfrac><mfrac><mn>1</mn><mrow><mi>ln</mi><mo stretchy="false">(</mo><msub><mi>c</mi><mn>0</mn></msub><mo>+</mo><msub><mi>c</mi><mn>1</mn></msub><mi>x</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>]</mo></mrow><mn>.</mn></mrow></math> In literature, there are other forms of the quark mass scaling, e.g., an isospin term was considered in Ref. [ 38 38 ]; Eqs. (1) and (2) were extended to finite temperature by expansion to a Taylor series of temperature [ 39 39 ]; The one-gluon-exchange effect was considered [ 40 40 ]; The asymptotic freedom was explicitly considered by a Wood-Saxon factor [ 41 41 ]; etc. In the present paper, however, we use the quark mass formula in Eq. (7) to study the properties of light-quark matter. 3 Properties of two-flavor quark matter 1 For simplicity, we assume the system consisting of light quarks and gluons at finite temperature T . According to the equivalent particle model, the free particle contribution to the thermodynamic potential density can be written in the form <math display="block" id="M15"><mrow><msub><mi>Ω</mi><mn>0</mn></msub><mo>=</mo><msubsup><mi>Ω</mi><mn>0</mn><mo>+</mo></msubsup><mo>+</mo><msubsup><mi>Ω</mi><mn>0</mn><mo>−</mo></msubsup><mo>+</mo><msubsup><mi>Ω</mi><mn>0</mn><mtext>g</mtext></msubsup><mn>,</mn></mrow></math> where the equivalent quark (+) and antiquark (-) contributions are <math display="block" id="M16"><mrow><msubsup><mi>Ω</mi><mn>0</mn><mo>±</mo></msubsup><mo>=</mo><mo>−</mo><mfrac><mrow><msub><mi>d</mi><mi>q</mi></msub><mi>T</mi></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mi>ln</mi><mrow><mo>[</mo><mrow><mn>1</mn><mo>+</mo><msup><mi>e</mi><mrow><mo>−</mo><mo stretchy="false">(</mo><msqrt><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt><mo>∓</mo><msup><mi>μ</mi><mo>*</mo></msup><mo stretchy="false">)</mo><mo>/</mo><mi>T</mi></mrow></msup></mrow><mo>]</mo></mrow><msup><mi>p</mi><mn>2</mn></msup><mtext> </mtext><mi>d</mi><mtext> </mtext><mi>p</mi><mn>,</mn></mrow></math> with the degeneracy factor d q =3(colors)× 2(spins)× 2(flavors)=12, while the contribution from gluons is <math display="block" id="M17"><mrow><msubsup><mi>Ω</mi><mn>0</mn><mtext>g</mtext></msubsup><mo>=</mo><mfrac><mrow><msub><mi>d</mi><mtext>g</mtext></msub><mi>T</mi></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mi>ln</mi><mrow><mo>[</mo><mrow><mn>1</mn><mo>−</mo><msup><mi>e</mi><mrow><mo>−</mo><msqrt><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mtext>g</mtext><mn>2</mn></msubsup></mrow></msqrt><mo>/</mo><mi>T</mi></mrow></msup></mrow><mo>]</mo></mrow><msup><mi>p</mi><mn>2</mn></msup><mtext> </mtext><mi>d</mi><mi>p</mi><mn>.</mn></mrow></math> The net quark baryon number density is then <math display="block" id="M18"><mrow><msub><mi>n</mi><mtext>b</mtext></msub><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mrow><mo>(</mo><mrow><msubsup><mi>n</mi><mi>q</mi><mo>+</mo></msubsup><mo>−</mo><msubsup><mi>n</mi><mi>q</mi><mo>−</mo></msubsup></mrow><mo>)</mo></mrow><mn>,</mn></mrow></math> where <math id="M19"><mrow><msubsup><mi>n</mi><mi>q</mi><mo>+</mo></msubsup></mrow></math> and <math id="M20"><mrow><msubsup><mi>n</mi><mi>q</mi><mo>−</mo></msubsup></mrow></math> are, respectively, the quark/antiquark number densities: <math display="block" id="M21"><mrow><msubsup><mi>n</mi><mi>q</mi><mo>±</mo></msubsup><mo>=</mo><mfrac><mrow><msub><mi>d</mi><mi>q</mi></msub></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mfrac><mrow><msup><mi>p</mi><mn>2</mn></msup><mtext>d</mtext><mi>p</mi></mrow><mrow><msup><mi>e</mi><mrow><mo stretchy="false">(</mo><msqrt><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt><mo>∓</mo><msup><mi>μ</mi><mo>*</mo></msup><mo stretchy="false">)</mo><mo>/</mo><mi>T</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac><mn>.</mn></mrow></math> We emphasize that the quark chemical potential becomes effective due to the density and temperature dependence of the quark mass. The effective chemical potential, μ * , is related to the actual chemical potential, μ , by <math display="block" id="M22"><mrow><mi>μ</mi><mo>=</mo><msup><mi>μ</mi><mo>*</mo></msup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mfrac><mrow><mo>∂</mo><msub><mi>Ω</mi><mn>0</mn></msub></mrow><mrow><mo>∂</mo><msub><mi>m</mi><mi>q</mi></msub></mrow></mfrac><mfrac><mrow><mo>∂</mo><msub><mi>m</mi><mi>q</mi></msub></mrow><mrow><mo>∂</mo><msub><mi>n</mi><mtext>b</mtext></msub></mrow></mfrac><mn>,</mn></mrow></math> where d m q /d n b is given in Eq. (11), and the derivative of Ω 0 , with respect to m q , is given by <math display="block" id="M23"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mo>∂</mo><msub><mi>Ω</mi><mn>0</mn></msub></mrow><mrow><mo>∂</mo><msub><mi>m</mi><mi>q</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>d</mi><mi>q</mi></msub><msub><mi>m</mi><mi>q</mi></msub></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mi>∞</mi></msubsup></mrow></mstyle><mrow><mo>[</mo><mrow><mfrac><mn>1</mn><mrow><msup><mi>e</mi><mrow><mo stretchy="false">(</mo><msqrt><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt><mo>−</mo><msup><mi>μ</mi><mo>*</mo></msup><mo stretchy="false">)</mo><mo>/</mo><mi>T</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mrow><mo>+</mo><mfrac><mn>1</mn><mrow><msup><mi>e</mi><mrow><mo stretchy="false">(</mo><msqrt><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt><mo>+</mo><msup><mi>μ</mi><mo>*</mo></msup><mo stretchy="false">)</mo><mo>/</mo><mi>T</mi></mrow></msup><mo>+</mo><mn>1</mn></mrow></mfrac></mrow><mo>]</mo></mrow><mfrac><mrow><msup><mi>p</mi><mn>2</mn></msup><mtext> </mtext><mi>d</mi><mtext> </mtext><mi>p</mi></mrow><mrow><msqrt><mrow><msup><mi>p</mi><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mi>q</mi><mn>2</mn></msubsup></mrow></msqrt></mrow></mfrac><mn>.</mn></mtd></mtr></mtable></math> The pressure of light-quark matter at finite temperature is <math display="block" id="M24"><mrow><mi>P</mi><mo>=</mo><mo>−</mo><msub><mi>Ω</mi><mn>0</mn></msub><mo>+</mo><msub><mi>n</mi><mtext>b</mtext></msub><mfrac><mrow><mo>∂</mo><msub><mi>m</mi><mi>q</mi></msub></mrow><mrow><mo>∂</mo><msub><mi>n</mi><mtext>b</mtext></msub></mrow></mfrac><mfrac><mrow><mo>∂</mo><msub><mi>Ω</mi><mn>0</mn></msub></mrow><mrow><mo>∂</mo><msub><mi>m</mi><mi>q</mi></msub></mrow></mfrac></mrow></math> with the second term on the right hand side of the above equation arising due to the density dependence of the quark mass. The entropy density reads <math display="block" id="M25"><mrow><mi>S</mi><mo>=</mo><mo>−</mo><mfrac><mrow><mo>∂</mo><msub><mi>Ω</mi><mn>0</mn></msub></mrow><mrow><mo>∂</mo><mi>T</mi></mrow></mfrac><mo>−</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mi>q</mi><mn>,</mn><mi>g</mi></mrow></munder></mstyle><mfrac><mrow><mo>∂</mo><msub><mi>m</mi><mi>i</mi></msub></mrow><mrow><mo>∂</mo><mi>T</mi></mrow></mfrac><mfrac><mrow><mo>∂</mo><msub><mi>Ω</mi><mn>0</mn></msub></mrow><mrow><mo>∂</mo><msub><mi>m</mi><mi>i</mi></msub></mrow></mfrac><mn>,</mn></mrow></math> while the free energy density is given by <math display="block" id="M26"><mrow><mi>F</mi><mo>=</mo><msub><mi>Ω</mi><mn>0</mn></msub><mo>+</mo><mn>3</mn><msub><mi>n</mi><mtext>b</mtext></msub><msup><mi>μ</mi><mo>*</mo></msup><mn>.</mn></mrow></math> Therefore, the energy density can be obtained by substituting Eqs. (23) and (24) into the thermodynamic relation E = F + TS , which leads to <math display="block" id="M27"><mrow><mi>E</mi><mo>=</mo><msub><mi>Ω</mi><mn>0</mn></msub><mo>+</mo><mn>3</mn><msub><mi>n</mi><mtext>b</mtext></msub><msup><mi>μ</mi><mo>*</mo></msup><mo>+</mo><mi>T</mi><mi>S</mi><mn>.</mn></mrow></math> For a given pair of the density, n b , and temperature, T , we can first solve Eq. (18) to obtain the effective chemical potential, μ * , and the thermodynamic properties can be calculated by Eqs. (22)–(25). Please note, in these expressions, the contribution from gluons has already been included. In Fig. 2 Fig. 2 , we show the energy per baryon as functions of the density at the fixed temperature values of T =100 MeV, 150 MeV, and 200 MeV for the parameters D 1/2 =160 MeV and C =0.6. At lower density, the energy per baryon becomes infinitely large, indicating quark confinement. At large baryon density, the energy per baryon also becomes large because the perturbative interactions become gradually important. The temperature dependence of the energy (dashed) and free energy (solid) per baryon are given in Fig. 3 Fig. 3 . We notice that the energy per baryon increases with temperature. However, the free energy per baryon decreases with increasing temperature, and eventually becomes negative. This is understandable with a view to the relation F = E - TS and the fact that the entropy term - TS dominates at large temperature. To check the thermodynamic consistency and demonstrate the effect of the perturbative interaction, we separately plot the density dependence of the energy (dashed curves) and free energy (solid) per baryon as a function of the density at the fixed temperature T =30 MeV for C =0 (open square) and C =0.6 (full square) in Fig. 4 Fig. 4 where the points marked with a triangle is the minimum while the points marked with an open circle are the zero pressure. In Fig. 5 Fig. 5 , the pressure and the corresponding free energy per baryon at T =50 MeV are simultaneously shown. It is clearly seen that the pressure is negative when the density is lower than the density where the free energy minimum. It is positive otherwise. It is also checked that the quantity E + P - TS -3 n b μ is zero at arbitrary density and temperature. In fact, combining Eqs. (20) and (22) one can easily get P =-Ω 0 +3 n b ( μ - μ * ), i.e., Ω 0 =- P +3 n b ( μ - μ * ). Then substituting it into Eq. (25) and combining similar terms, one immediately have <math display="block" id="M28"><mrow><mi>E</mi><mo>+</mo><mi>P</mi><mo>−</mo><mi>T</mi><mi>S</mi><mo>−</mo><mn>3</mn><msub><mi>n</mi><mtext>b</mtext></msub><mi>μ</mi><mo>=</mo><mn>0.</mn></mrow></math> 4 Summary 1 We have studied the properties of light-quark matter in the equivparticle model with a new quark mass scaling at finite density and temperature. Due to the density and temperature dependence of the quark matter, the chemical potential becomes effective, which ensure the thermodynamic consistency: the pressure at the minimum free energy per baryon is exactly zero. We extend the quark mass scaling with both quark confinement and first-order perturbative interactions to finite temperature, and apply it to the investigation of light-quark matter. With increasing temperature, the energy per baryon increases while the free energy per baryon decreases. The perturbative interaction increases with density and temperature. It should be pointed out that the present studies are limited to two-flavor quark matter. Extension to the real three flavor quark matter is a meaningful task, and will be done in the near future. References J.F. Xu , G.X. Peng , F. Liu , et al . Strange matter and strange stars in a thermodynamically self-consistent perturbation model with running coupling and running strange quark mass . Phys Rev D , 2015 , 92 : 025025 . DOI: 10.1103/PhysRevD.92.025025 http://doi.org/10.1103/PhysRevD.92.025025 Strange matter and strange stars in a thermodynamically self-consistent perturbation model with running coupling and running strange quark mass Y. Nambu , G. Jona-Lasino . Dynamical model of elementary particles based on an analogy with superconductivity. I . Phys. Rev , 1961 , 122 : 345 - 358 . DOI: 10.1103/PhysRev.122.345 http://doi.org/10.1103/PhysRev.122.345 Dynamical model of elementary particles based on an analogy with superconductivity. I E.S. Fraga , R.D. Pisarski , J. Schaffner-Bielich . Small, dense quark stars from perturbative QCD . Phys Rev D , 2001 , 63 : 121702(R) . DOI: 10.1103/PhysRevD.63.121702 http://doi.org/10.1103/PhysRevD.63.121702 Small, dense quark stars from perturbative QCD G.X. Peng . Thermodynamic correction to the strong interaction in the perturbative regime . Europhys Lett , 2005 , 72 : 69 - 75 . DOI: 10.1209/epl/i2005-10189-8 http://doi.org/10.1209/epl/i2005-10189-8 Thermodynamic correction to the strong interaction in the perturbative regime J.F. Xu , G.X. Peng , Z.Y. Lu , et al . Two-flavor quark matter in the perturbation theory with full thermodynamic consistency . Sci China-Phys Mech Astron , 2015 , 58 : 042001 . DOI: 10.1007/s11433-014-5599-6 http://doi.org/10.1007/s11433-014-5599-6 Two-flavor quark matter in the perturbation theory with full thermodynamic consistency S. Plumari , G.F. Bugio , V. Greco V . Quark matter in neutron stars within the field correlator method . Phys Rev D , 2013 , 88 : 083005 . DOI: 10.1103/PhysRevD.88.083005 http://doi.org/10.1103/PhysRevD.88.083005 Quark matter in neutron stars within the field correlator method R.X. Xu . Can the age discrepancies of neutron stars be circumvented by an accretion-assisted torque? Astrophys J , 2003 , 596 , L75 - L78 ; Can the age discrepancies of neutron stars be circumvented by an accretion-assisted torque? Can cold quark matter be solid? Int J Mod Phys D , 2010 , 19 : 1437 - 1446 . DOI: 10.1142/S0218271810017767 http://doi.org/10.1142/S0218271810017767 Can cold quark matter be solid? G.S. Khadekar , R. Wanjari . Geometry of quark and strange quark matter in higher dimensional general relativity . Int J Theor Phys , 2012 , 51 : 1408 - 1415 . DOI: 10.1007/s10773-011-1016-3 http://doi.org/10.1007/s10773-011-1016-3 Geometry of quark and strange quark matter in higher dimensional general relativity P.K. Sahoo , B. Mishra . Axially symmetric space-time with strange quark matter attached to string cloud in bimetric theory . Int J Pure Appl Math , 2013 , 82 : 87 - 94 . http://ijpam.eu/contents/2013-82-1/8/ http://ijpam.eu/contents/2013-82-1/8/ Axially symmetric space-time with strange quark matter attached to string cloud in bimetric theory X.J. Wen . Color-flavor locked strange quark matter in a strong magnetic field . Phys Rev D , 2013 , 88 : 034031 . DOI: 10.1103/PhysRevD.88.034031 http://doi.org/10.1103/PhysRevD.88.034031 Color-flavor locked strange quark matter in a strong magnetic field A.A. Isayev , J.J. Yang . Anisotropic pressure in strange quark matter in the presence of a strong magnetic field . J Phys G Nucl Part Phys , 2013 , 40 : 035105 . DOI: 10.1088/0954-3899/40/3/035105 http://doi.org/10.1088/0954-3899/40/3/035105 Anisotropic pressure in strange quark matter in the presence of a strong magnetic field X.Y. Wang , I.A. Shovkovy . Bulk viscosity of spin-one color superconducting strange quark matter . Phys Rev D , 2010 , 82 : 085007 ; DOI: 10.1103/PhysRevD.82.085007 http://doi.org/10.1103/PhysRevD.82.085007 Bulk viscosity of spin-one color superconducting strange quark matter M. Huang , I.A. Shovkovy . Chromomagnetic instability in dense quark matter . Phys Rev D , 2004 , 70 : 051501(R) . DOI: 10.1103/PhysRevD.70.051501 http://doi.org/10.1103/PhysRevD.70.051501 Chromomagnetic instability in dense quark matter T. Bao , G.Z. Liu , E.G. Zhao , et al . Self-consistently thermodynamic treatment for strange quark matter in the effective mass bag model . Eur Phys J A , 2008 , 38 : 287 - 293 . DOI: 10.1140/epja/i2008-10682-6 http://doi.org/10.1140/epja/i2008-10682-6 Self-consistently thermodynamic treatment for strange quark matter in the effective mass bag model G.X. Peng , H.C. Chiang , J.J. Yang , et al . Thermodynamics, strange quark matter, and strange stars . Phys Rev C , 2000 , 61 : 015201 . DOI: 10.1103/PhysRevC.62.025801 http://doi.org/10.1103/PhysRevC.62.025801 Thermodynamics, strange quark matter, and strange stars G.X. Peng , H.C. Chiang , P.Z. Ning , et al . Charge and critical density of strange quark matter . Phys Rev C , 1999 , 59 : 3452 - 3454 . DOI: 10.1103/PhysRevC.59.3452 http://doi.org/10.1103/PhysRevC.59.3452 Charge and critical density of strange quark matter X.J. Wen , X.H. Zhong , G.X. Peng , et al . Thermodynamics with density and temperature dependent particle masses and properties of bulk strange quark matter and strangelets . Phys Rev C , 2005 , 72 : 015204 . DOI: 10.1103/PhysRevC.72.015204 http://doi.org/10.1103/PhysRevC.72.015204 Thermodynamics with density and temperature dependent particle masses and properties of bulk strange quark matter and strangelets J.X. Hou , G.X. Peng , C.J. Xia , et al . Magnetized strange quark matter in a mass-density-dependent model . Chin Phys C , 2015 , 39 : 015101 . DOI: 10.1088/1674-1137/39/1/015101 http://doi.org/10.1088/1674-1137/39/1/015101 Magnetized strange quark matter in a mass-density-dependent model S.S. Cui , G.X. Peng , Z.Y. Lu , et al . Properties of color-flavor locked strange quark matter in an external strong magnetic field . Nucl Sci Tech , 2015 , 26 : 040503 . DOI: 10.13538/j.1001-8042/nst.26.040503 http://doi.org/10.13538/j.1001-8042/nst.26.040503 Properties of color-flavor locked strange quark matter in an external strong magnetic field V. Goloviznin , H. Satz . The refractive properties of the gluon plasma in SU(2) gauge theory . Z Phys C , 1993 , 57 : 671 - 675 . DOI: 10.1007/BF01561487 http://doi.org/10.1007/BF01561487 The refractive properties of the gluon plasma in SU(2) gauge theory A. Peshier , B. Kämpfer , O.P. Pavlenko , et al . An effective model of the quark-gluon plasma with thermal parton masses . In: Letessier L , et al, editors. Hot Hadronic Matter . New York : Plenum Press , 1995 , 139 - 142 . An effective model of the quark-gluon plasma with thermal parton masses M.I. Gorenstein , S.N. Yang . Gluon plasma with a medium-dependent dispersion relation . Phys Rev D , 1995 , 52 : 5206 - 5212 . DOI: 10.1103/PhysRevD.52.5206 http://doi.org/10.1103/PhysRevD.52.5206 Gluon plasma with a medium-dependent dispersion relation M. Bluhm , B. Kämpfer , G. Soff . The QCD equation of state near Tc within a quasi-particle model . Phys Lett B , 2005 , 620 : 131 - 136 . DOI: 10.1016/j.physletb.2005.05.083 http://doi.org/10.1016/j.physletb.2005.05.083 The QCD equation of state near Tc within a quasi-particle model V.M. Bannur . Self-consistent quasiparticle model for quark-gluon plasma . Phys Rev C , 2007 , 75 : 044905 . DOI: 10.1103/PhysRevC.75.044905 http://doi.org/10.1103/PhysRevC.75.044905 Self-consistent quasiparticle model for quark-gluon plasma F.G. Gardim , F.M. Steffens . Thermodynamics of quasi-particles at finite chemical potential . Nucl Phys A , 2009 , 825 : 222 - 244 . DOI: 10.1016/j.nuclphysa.2009.05.001 http://doi.org/10.1016/j.nuclphysa.2009.05.001 Thermodynamics of quasi-particles at finite chemical potential H. Li , X.L. Luo , H.S. Zong . Bag model and quark star . Phys Rev D , 2010 , 82 : 065017 . DOI: 10.1103/PhysRevD.82.065017 http://doi.org/10.1103/PhysRevD.82.065017 Bag model and quark star L.J. Luo , J.C. Yan , W.M. Sun , et al . A thermodynamically consistent quasi-particle model without density-dependent infinity of the vacuum zero-point energy . Eur Phys J C , 2013 , 73 : 2626 . DOI: 10.1140/epjc/s10052-013-2626-0 http://doi.org/10.1140/epjc/s10052-013-2626-0 A thermodynamically consistent quasi-particle model without density-dependent infinity of the vacuum zero-point energy C. Wu , W.L. Qian , R.K. Su . Improved quark mass density-dependent model with quark and nonlinear scalar field coupling . Phys Rev C , 2005 , 72 : 035205 . DOI: 10.1103/PhysRevC.72.035205 http://doi.org/10.1103/PhysRevC.72.035205 Improved quark mass density-dependent model with quark and nonlinear scalar field coupling Z.Y. Lu , G.X. Peng , J.F. Xu , et al . Properties of quark matter in a new quasiparticle model with QCD running coupling . Sci China-Phys Mech Astron , 2016 , 59 : 662001 . DOI: 10.1007/s11433-015-0524-2 http://doi.org/10.1007/s11433-015-0524-2 Properties of quark matter in a new quasiparticle model with QCD running coupling G.N. Fowler , S. Raha , R.M. Weiner . Confinement and phase transitions . Z Phys C , 1981 , 9 : 271 . DOI: 10.1007/BF01410668 http://doi.org/10.1007/BF01410668 Confinement and phase transitions S. Chakrabarty , S. Raha , B. Sinha . Strange quark matter and the mechanism of confinement . Phys Lett B , 1989 , 229 : 112 - 116 . DOI: 10.1016/0370-2693(89)90166-4 http://doi.org/10.1016/0370-2693(89)90166-4 Strange quark matter and the mechanism of confinement O.G. Benvenuto , G. Lugones . Strange matter equation of state in the quark mass-density-dependent model . Phys Rev D , 1995 , 51 : 1989 . DOI: 10.1103/PhysRevD.51.1989 http://doi.org/10.1103/PhysRevD.51.1989 ; Strange matter equation of state in the quark mass-density-dependent model G. Lugones , O.G. Benvenuto . Strange matter equation of state and the combustion of nuclear matter into strange matter in the quark mass-density-dependent model at T>0 . Phys Rev D , 1995 , 52 : 1276 . DOI: 10.1103/PhysRevD.52.1276 http://doi.org/10.1103/PhysRevD.52.1276 Strange matter equation of state and the combustion of nuclear matter into strange matter in the quark mass-density-dependent model at T>0 G.X. Peng , H.C. Chiang , B.S. Zhou , et al . Thermodynamics, strange quark matter, and strange stars . Phys Rev C , 2000 , 62 : 025801 . DOI: 10.1103/PhysRevC.62.025801 http://doi.org/10.1103/PhysRevC.62.025801 Thermodynamics, strange quark matter, and strange stars G.X. Peng , A. Li , U. Lombardo . Deconfinement phase transition in hybrid neutron stars from the Brueckner theory with three-body forces and a quark model with chiral mass scaling . Phys Rev C , 2008 , 77 : 065807 . DOI: 10.1103/PhysRevC.77.065807 http://doi.org/10.1103/PhysRevC.77.065807 Deconfinement phase transition in hybrid neutron stars from the Brueckner theory with three-body forces and a quark model with chiral mass scaling C.J. Xia , G.X. Peng , S.W. Chen , et al . Thermodynamic consistency, quark mass scaling, and properties of strange matter . Phys Rev D , 2014 , 89 : 105027 . DOI: 10.1103/PhysRevD.89.105027 http://doi.org/10.1103/PhysRevD.89.105027 Thermodynamic consistency, quark mass scaling, and properties of strange matter G.X. Peng , X.J. Wen , Y.D. Chen . New solutions for the color-flavor locked strangelets . Phys Lett B , 2006 , 633 : 314 - 318 . DOI: 10.1016/j.physletb.2006.01.032 http://doi.org/10.1016/j.physletb.2006.01.032 New solutions for the color-flavor locked strangelets X.J. Wen , G.X. Peng , Y.D. Chen . Charge, strangeness and radius of strangelets . J Phys G Nucl Part Phys , 2007 , 34 : 1697 . DOI: 10.1088/0954-3899/34/7/010 http://doi.org/10.1088/0954-3899/34/7/010 Charge, strangeness and radius of strangelets X.P. Zheng , X.W. Liu , M. Kang , et al . Bulk viscosity of strange quark matter in a density-dependent quark mass model and dissipation of the r mode in strange stars . Phys Rev C , 2004 , 70 : 015803 . DOI: 10.1103/PhysRevC.70.015803 http://doi.org/10.1103/PhysRevC.70.015803 Bulk viscosity of strange quark matter in a density-dependent quark mass model and dissipation of the r mode in strange stars G. Lugones , J.E. Horvath . Quark-diquark equation of state and compact star structure . Int J Mod Phys D , 2003 , 12 : 495 . DOI: 10.1142/S0218271803002755 http://doi.org/10.1142/S0218271803002755 Quark-diquark equation of state and compact star structure P.C. Chu , L.W. Chen . Quark matter symmetry energy and quark stars . Astrophys J , 2014 , 780 : 135 . DOI: 10.1088/0004-637X/780/2/135 http://doi.org/10.1088/0004-637X/780/2/135 Quark matter symmetry energy and quark stars Y. Zhang , R.K. Su . Quark mass density- and temperature-dependent model for bulk strange quark matter . Phys Rev C , 2013 , 65 , 035202 . DOI: 10.1103/PhysRevC.65.035202 http://doi.org/10.1103/PhysRevC.65.035202 Quark mass density- and temperature-dependent model for bulk strange quark matter C.J. Xia , S.W. Chen , G.X. Peng . Properties of strangelets in a new quark mass confinement model with one-gluon-exchange interaction . Sci China-Phys Mech Astron , 2014 , 57 : 1304 - 1310 . DOI: 10.1007/s11433-014-5452-y http://doi.org/10.1007/s11433-014-5452-y Properties of strangelets in a new quark mass confinement model with one-gluon-exchange interaction C. Peng , G.X. Peng , C.J. Xia , et al . Magnetized strange quark matter in the equivparticle model with both confinement and perturbative interactions . Nucl Sci Tech , 2016 , 27 : 98 . DOI: 10.1007/s41365-016-0095-5 http://doi.org/10.1007/s41365-016-0095-5 Magnetized strange quark matter in the equivparticle model with both confinement and perturbative interactions G.X. Peng . Renormalization group dependence of the QCD coupling . Phys Lett B , 2006 , 634 : 413 . DOI: 10.1016/j.physletb.2006.01.032 http://doi.org/10.1016/j.physletb.2006.01.032 Renormalization group dependence of the QCD coupling 10.1007/s41365-016-0148-9 This work was supported by the National Natural Science Foundation of China (Nos. 11135011, 11475110, and 11575190), and the CAS Present’s International Fellowship Initiative (Nos. 2015PM008 and 2016VMA063) . 2016-09-07 2016-10-03 2016-10-09 2016-10-31 2016-12-20 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2016 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-016-0148-9 http://dx.doi.org/10.1007/s41365-016-0148-9 . 2016 Nuclear Science and Techniques 06 27