Asymmetric neutron stars in the presence of vacuum fluctuations

Asymmetric neutron stars in the presence of vacuum fluctuations 1 first-author 2 1 3 corresp Corresponding author, jfyang@phy.ecnu.edu.cn Corresponding author, jfyang@phy.ecnu.edu.cn jfyang@phy.ecnu.edu.cn Department of Physics, East China Normal University, Shanghai 200241, China Department of Physics, East China Normal University, Shanghai 200241, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China Kavli Institute for Theoretical Physics China, Chinese Academy of Sciences, Beijing 100190, China Kavli Institute for Theoretical Physics China, Chinese Academy of Sciences, Beijing 100190, China The asymmetric neutron stars are investigated in a relativistic effective model with vacuum fluctuations (VF) taken into account. Due to the VF effects, various properties of the neutron matter become ’softened’ comparing to that obtained in the FSUGold model, and the maximum mass of the ( npeμ ) neutron stars is reduced from 1.71M_⊙ to 1.35M ⊙ . Neutron star Relativistic mean-field theory Vacuum fluctuation Equation of state 1 Introduction 1 The neutron stars with masses of the order of solar mass are the result of supernova explosions. As their radii are merely 10–12 km [ 1 1 ], the interior density of a neutron star exceeds several times the nuclear saturation density. Thus, neutron star matter provides an interesting laboratory for studying strong interactions at a high density of nucleons and has been attracting a lot of academic interest [ 2 2 - 4 4 2-4 ]. In this work, we will concentrate our investigation on the asymmetric neutron stars consisting of neutrons, proton, and leptons. The basic framework to be employed is the relativistic mean field theory (RMF) that has proven successful for the study of nuclear matter as a relativistic many-body system of baryons and mesons [ 5 5 , 6 6 ]. Ever since the seminal work of the Walecka model [ 5 5 ], there has been many important extensions of quantum hadron dynamics (QHD) that improved our understanding of hadron matter at a high density. For example, an important theoretical advancement appeared in early 2001 [ 7 7 ], where nonlinear couplings between the isovector and isoscalar mesons was introduced to soften the symmetry energy of nuclear matter at a high density using an appropriate data set called the FSUGold model [ 8 8 ]. Further applications of this model and its extensions proved to be successful in many aspects [ 9 9 , 10 10 ]. As noted in Ref. [ 11 11 ], it is better to use the in-medium the meson masses in such RMF studies of nuclear matter at a high density. (Actually such in-medium meson masses have been studied in both experimental and theoretical approaches [ 12 12 - 17 17 12-17 ].) Following Refs. [ 11 11 , 18 18 ], we will calculate the effective masses of nucleons. Mesons are calculated by taking into account the effects of the vacuum fluctuation (VF) in the extension of QHD proposed in Refs. [ 7 7 , 8 8 ]. We will also investigate how the properties of neutron matter are affected. Our results w ill be compared to these obtained in the FSUGold model [ 19 19 , 20 20 ]. This work is organised as follows: In Sec. 2, we present the framework (henceforth referred to as VF-RMF model as in Ref. [ 18 18 ]) and theoretical results for our investigation of various properties for neutron stars. In Sec. 3, the numerical results are presented in comparison with that of FSUGold model. The summary is given in Sec. 4. 2 Effective masses and nuclear matter 1 The Lagrangian density employed in this work reads [ 20 20 ] http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339372&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339376&type= where N represents nucleons n and p , and l represents leptons e - and μ - , m N , m , m ω and m ρ denote the masses for N and , ω , ρ , respectively. F μν and <math id="M1"><mrow><msub><mover accent="true"><mi>G</mi><mo>→</mo></mover><mrow><mi>μ</mi><mi>ν</mi></mrow></msub></mrow></math> denote the antisymmetric tensors of vector fields ω and ρ , respectively [ 20 20 ]. Note that Λ v is the coupling first introduced in Ref. [ 7 7 ]. The meson field equations in the RMF read: <math display="block" id="M2"><mrow><msubsup><mi>m</mi><mi>σ</mi><mn>2</mn></msubsup><mi>σ</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>κ</mi><msubsup><mi>g</mi><mrow><mi>σ</mi><mi>N</mi></mrow><mn>3</mn></msubsup><msup><mi>σ</mi><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mn>6</mn></mfrac><mi>λ</mi><msubsup><mi>g</mi><mrow><mi>σ</mi><mi>N</mi></mrow><mn>4</mn></msubsup><msup><mi>σ</mi><mn>3</mn></msup><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>N</mi></munder></mstyle><msub><mi>g</mi><mrow><mi>σ</mi><mi>N</mi></mrow></msub><msubsup><mi>ρ</mi><mi>N</mi><mi>S</mi></msubsup><mn>,</mn></mrow></math> <math display="block" id="M3"><mrow><msubsup><mi>m</mi><mi>ω</mi><mn>2</mn></msubsup><mi>ω</mi><mo>+</mo><mfrac><mi>ζ</mi><mn>6</mn></mfrac><msubsup><mi>g</mi><mrow><mi>ω</mi><mi>N</mi></mrow><mn>4</mn></msubsup><msup><mi>ω</mi><mn>3</mn></msup><mo>+</mo><mn>2</mn><msub><mi>Λ</mi><mi>v</mi></msub><msubsup><mi>g</mi><mrow><mi>ρ</mi><mi>N</mi></mrow><mn>2</mn></msubsup><msubsup><mi>g</mi><mrow><mi>ω</mi><mi>N</mi></mrow><mn>2</mn></msubsup><msup><mi>ρ</mi><mn>2</mn></msup><mi>ω</mi><mo>=</mo><mstyle displaystyle="true"><munder><mo>∑</mo><mi>N</mi></munder></mstyle><msub><mi>g</mi><mrow><mi>ω</mi><mi>N</mi></mrow></msub><msub><mi>ρ</mi><mi>N</mi></msub><mn>,</mn></mrow></math> <math display="block" id="M4"><mrow><msubsup><mi>m</mi><mi>ρ</mi><mn>2</mn></msubsup><mi>ρ</mi><mo>+</mo><mn>2</mn><msub><mi>Λ</mi><mi>v</mi></msub><msubsup><mi>g</mi><mrow><mi>ρ</mi><mi>N</mi></mrow><mn>2</mn></msubsup><msubsup><mi>g</mi><mrow><mi>ω</mi><mi>N</mi></mrow><mn>2</mn></msubsup><msup><mi>ω</mi><mn>2</mn></msup><mi>ρ</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>N</mi></munder></mstyle><msub><mi>g</mi><mrow><mi>ρ</mi><mi>N</mi></mrow></msub><msub><mi>τ</mi><mrow><mn>3</mn><mi>N</mi></mrow></msub><msub><mi>ρ</mi><mi>N</mi></msub><mn>,</mn></mrow></math> where ρ N and <math id="M5"><mrow><msubsup><mi>ρ</mi><mi>N</mi><mi>S</mi></msubsup></mrow></math> are the nucleon density and scalar density, respectively, <math display="block" id="M6"><mrow><msub><mi>ρ</mi><mi>N</mi></msub><mo>=</mo><mn>2</mn><mstyle displaystyle="true"><mrow><msubsup><mo>∫</mo><mn>0</mn><mrow><msubsup><mi>k</mi><mi>F</mi><mi>N</mi></msubsup></mrow></msubsup></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mi>k</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M7"><mrow><msubsup><mi>ρ</mi><mi>N</mi><mi>S</mi></msubsup><mo>=</mo><mo>−</mo><mi>i</mi><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>4</mn></msup><mi>k</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow></mfrac><mtext>Tr</mtext><mo stretchy="false">[</mo><msubsup><mi>G</mi><mi>D</mi><mi>N</mi></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mn>,</mn></mrow></math> with <math id="M8"><mrow><msubsup><mi>k</mi><mi>F</mi><mi>N</mi></msubsup></mrow></math> being the Fermi momentum of the nucleon and <math id="M9"><mrow><msubsup><mi>G</mi><mi>D</mi><mi>N</mi></msubsup></mrow></math> being the in-medium part of the propagator for quasi-nucleons <math display="block" id="M10"><mrow><msup><mi>G</mi><mi>N</mi></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≡</mo><msubsup><mi>G</mi><mi>F</mi><mi>N</mi></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>+</mo><msubsup><mi>G</mi><mi>D</mi><mi>N</mi></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M11"><mrow><msubsup><mi>G</mi><mi>F</mi><mi>N</mi></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≡</mo><mfrac><mrow><msup><mi>γ</mi><mi>μ</mi></msup><msub><mi>k</mi><mi>μ</mi></msub><mo>+</mo><msubsup><mi>m</mi><mi>N</mi><mo>*</mo></msubsup></mrow><mrow><msup><mi>k</mi><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>m</mi><mi>N</mi><mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><mi>i</mi><mi>ϵ</mi></mrow></mfrac><mn>,</mn></mrow></math> <math display="block" id="M12"><mrow><msubsup><mi>G</mi><mi>D</mi><mi>N</mi></msubsup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>≡</mo><mi>i</mi><mi>π</mi><mfrac><mrow><msup><mi>γ</mi><mi>μ</mi></msup><msub><mi>k</mi><mi>μ</mi></msub><mo>+</mo><msubsup><mi>m</mi><mi>N</mi><mo>*</mo></msubsup></mrow><mrow><msubsup><mi>E</mi><mi>F</mi><mi>N</mi></msubsup></mrow></mfrac><mi>δ</mi><mo stretchy="false">(</mo><msup><mi>k</mi><mn>0</mn></msup><mo>−</mo><msubsup><mi>E</mi><mi>F</mi><mi>N</mi></msubsup><mo stretchy="false">)</mo><mi>θ</mi><mo stretchy="false">(</mo><msubsup><mi>k</mi><mi>F</mi><mi>N</mi></msubsup><mo>−</mo><mo>|</mo><mover accent="true"><mi>k</mi><mo>→</mo></mover><mo>|</mo><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where <math id="M13"><mrow><msubsup><mi>E</mi><mi>F</mi><mi>N</mi></msubsup><mo>≡</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>k</mi><mi>F</mi><mi>N</mi></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>m</mi><mi>N</mi><mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt></mrow></math> and <math id="M14"><mrow><msubsup><mi>m</mi><mi>N</mi><mo>*</mo></msubsup></mrow></math> is the effective mass that includes the VF effects as below ( Fig. 1(a) Fig. 1(a) )[ 11 11 , 18 18 ]: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339398&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339394&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339398&type= The effective (in-medium) meson masses are calculated in the random phase approximation (RPA) ( Fig. 1(b) Fig. 1(b) ) [ 21 21 ], <math display="block" id="M15"><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>m</mi><mi>σ</mi><mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><msubsup><mi>m</mi><mi>σ</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>Π</mi><mi>σ</mi></msub><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> <math display="block" id="M16"><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>m</mi><mi>i</mi><mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>=</mo><msubsup><mi>m</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>Π</mi><mrow><mi>i</mi><mn>,</mn><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mn>,</mn><mtext> </mtext><mo stretchy="false">(</mo><mi>i</mi><mo>=</mo><mi>ω</mi><mn>,</mn><mi>ρ</mi><mo stretchy="false">)</mo></mrow></math> with ∏ i , T being the corresponding transverse part of the polarization tensor of vector mesons, <math display="block" id="M17"><mrow><msub><mi>Π</mi><mi>σ</mi></msub><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>i</mi><msubsup><mi>g</mi><mrow><mi>σ</mi><mi>N</mi></mrow><mn>2</mn></msubsup><mstyle displaystyle="true"><munder><mo>∑</mo><mi>N</mi></munder></mstyle><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>4</mn></msup><mi>k</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow></mfrac><mtext>Tr</mtext><mrow><mo>[</mo><mrow><msup><mi>G</mi><mi>N</mi></msup><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mi>q</mi><mo stretchy="false">)</mo><msup><mi>G</mi><mi>N</mi></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M18"><mrow><msub><mi>Π</mi><mrow><mi>ω</mi><mn>,</mn><mi>μ</mi><mi>ν</mi></mrow></msub><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>i</mi><msubsup><mi>g</mi><mrow><mi>ω</mi><mi>N</mi></mrow><mn>2</mn></msubsup><mstyle displaystyle="true"><munder><mo>∑</mo><mi>N</mi></munder></mstyle><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>4</mn></msup><mi>k</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow></mfrac><mtext>Tr</mtext><mrow><mo>[</mo><mrow><msub><mi>γ</mi><mi>μ</mi></msub><msup><mi>G</mi><mi>N</mi></msup><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mi>q</mi><mo stretchy="false">)</mo><msub><mi>γ</mi><mi>ν</mi></msub><msup><mi>G</mi><mi>N</mi></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M19"><mrow><msub><mi>Π</mi><mrow><mi>ρ</mi><mn>,</mn><mi>μ</mi><mi>ν</mi></mrow></msub><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>i</mi><mfrac><mrow><msubsup><mi>g</mi><mrow><mi>ρ</mi><mi>N</mi></mrow><mn>2</mn></msubsup></mrow><mn>4</mn></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>N</mi></munder></mstyle><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>4</mn></msup><mi>k</mi></mrow><mrow><msup><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mo stretchy="false">)</mo></mrow><mn>4</mn></msup></mrow></mfrac><mtext>Tr</mtext><mrow><mo>[</mo><mrow><msub><mi>τ</mi><mi>N</mi></msub><msub><mi>γ</mi><mi>μ</mi></msub><msup><mi>G</mi><mi>N</mi></msup><mo stretchy="false">(</mo><mi>k</mi><mo>+</mo><mi>q</mi><mo stretchy="false">)</mo><msub><mi>τ</mi><mi>N</mi></msub><msub><mi>γ</mi><mi>ν</mi></msub><msup><mi>G</mi><mi>N</mi></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> with which one could obtain on-shell and off-shell in-medium meson masses. The expressions of the off-shell ( <math id="M20"><mrow><msup><mi>q</mi><mi>μ</mi></msup><mo>=</mo><mn>0</mn></mrow></math> ) case are quite simple: <math display="block" id="M21"><mrow><msub><mi>Π</mi><mi>σ</mi></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mi>μ</mi></msup><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn>3</mn><msubsup><mi>g</mi><mrow><mi>σ</mi><mi>N</mi></mrow><mn>2</mn></msubsup></mrow><mrow><mn>2</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>N</mi></munder></mstyle><mrow><mo>[</mo><mrow><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>m</mi><mi>N</mi><mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><msubsup><mi>k</mi><mi>F</mi><mi>N</mi></msubsup><mo>+</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>k</mi><mi>F</mi><mi>N</mi></msubsup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow><mrow><msubsup><mi>E</mi><mi>F</mi><mi>N</mi></msubsup></mrow></mfrac><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>m</mi><mi>N</mi><mo>*</mo></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mi>ln</mi><mfrac><mrow><msubsup><mi>k</mi><mi>F</mi><mi>N</mi></msubsup><mo>+</mo><msubsup><mi>E</mi><mi>F</mi><mi>N</mi></msubsup></mrow><mrow><msub><mi>m</mi><mi>N</mi></msub></mrow></mfrac><mo>+</mo><mfrac><mrow><msubsup><mi>m</mi><mi>N</mi><mn>2</mn></msubsup><mo>+</mo><mn>3</mn><mo stretchy="false">(</mo><msubsup><mi>m</mi><mi>N</mi><mo>*</mo></msubsup><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow><mn>2</mn></mfrac><mo>−</mo><mn>2</mn><msubsup><mi>m</mi><mi>N</mi><mo>*</mo></msubsup><msub><mi>m</mi><mi>N</mi></msub></mrow><mo>]</mo></mrow><mn>,</mn></mrow></math> <math display="block" id="M22"><mrow><msub><mi>Π</mi><mrow><mi>ω</mi><mn>,</mn><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mi>μ</mi></msup><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mi>g</mi><mrow><mi>ω</mi><mi>N</mi></mrow><mn>2</mn></msubsup></mrow><mrow><mn>3</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>N</mi></munder></mstyle><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>k</mi><mi>F</mi><mi>N</mi></msubsup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow><mrow><msubsup><mi>E</mi><mi>F</mi><mi>N</mi></msubsup></mrow></mfrac><mn>,</mn><mtext> </mtext><msub><mi>Π</mi><mrow><mi>ρ</mi><mn>,</mn><mi>T</mi></mrow></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mi>μ</mi></msup><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msubsup><mi>g</mi><mrow><mi>ρ</mi><mi>N</mi></mrow><mn>2</mn></msubsup></mrow><mrow><mn>24</mn><msup><mi>π</mi><mn>2</mn></msup></mrow></mfrac><mstyle displaystyle="true"><munder><mo>∑</mo><mi>N</mi></munder></mstyle><mfrac><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>k</mi><mi>F</mi><mi>N</mi></msubsup><mo stretchy="false">)</mo></mrow><mn>3</mn></msup></mrow><mrow><msubsup><mi>E</mi><mi>F</mi><mi>N</mi></msubsup></mrow></mfrac><mn>.</mn></mrow></math> For the asymmetric neutron star matter with nucleons and charged leptons, the β equilibrium conditions are guaranteed with the following relations of chemical potentials <math display="block" id="M23"><mrow><msub><mi>μ</mi><mi>p</mi></msub><mo>=</mo><msub><mi>μ</mi><mi>n</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>e</mi></msub><mn>,</mn><mtext> </mtext><msub><mi>μ</mi><mi>μ</mi></msub><mo>=</mo><msub><mi>μ</mi><mi>e</mi></msub><mn>,</mn></mrow></math> and the charge neutrality condition <math display="block" id="M24"><mrow><msub><mi>n</mi><mi>p</mi></msub><mo>=</mo><msub><mi>n</mi><mi>e</mi></msub><mo>+</mo><msub><mi>n</mi><mrow><msup><mi>μ</mi><mo>−</mo></msup></mrow></msub><mn>,</mn></mrow></math> where n i is the number density of particle i , and the chemical potentials of nucleons and leptons read <math display="block" id="M25"><mrow><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>μ</mi><mi>N</mi></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msubsup><mi>E</mi><mi>F</mi><mi>N</mi></msubsup><mo>+</mo><msub><mi>g</mi><mrow><mi>ω</mi><mi>N</mi></mrow></msub><mi>ω</mi><mo>+</mo><msub><mi>g</mi><mrow><mi>ρ</mi><mi>N</mi></mrow></msub><msub><mi>τ</mi><mrow><mn>3</mn><mi>N</mi></mrow></msub><mi>ρ</mi><mn>,</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><msub><mi>μ</mi><mi>l</mi></msub></mrow></mtd><mtd columnalign="left"><mrow><mo>=</mo><msqrt><mrow><msup><mrow><mo stretchy="false">(</mo><msubsup><mi>k</mi><mi>F</mi><mi>l</mi></msubsup><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msubsup><mi>m</mi><mi>l</mi><mn>2</mn></msubsup></mrow></msqrt><mn>,</mn></mrow></mtd></mtr></mtable></mrow></math> where <math id="M26"><mrow><msubsup><mi>k</mi><mi>F</mi><mi>l</mi></msubsup></mrow></math> denotes the Fermi momentum of a lepton. Then, the equations of state (EOS) obtained in this VF-RMF model read, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339408&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339427&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339423&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=10339427&type= With these EOS, the mass-radius relation, other properties of neutron stars can be computed by solving the Tolman-Oppenheimer-Volkoff (TOV) equation [ 1 1 ]. 3 Numerical results and discussions 1 The nucleon and meson masses and the isoscalar couplings of the FSUGold data set, listed in Table 1 Table 1 , are adopted in our calculation. The couplings ( g σN , g ωN , g ρN ) are determined so that the following saturation properties are reproduced: nucleon density ρ 0 =0.148fm -3 , binding energy per nucleon E/A =-16.3 MeV, and the symmetry energy E sym =32.5 MeV, they are listed in Table 2 Table 2 together with the FSUGold data set for comparison. 1 1 Parameters 1 1 Values 1 1 m (MeV) 1 1 491.5 1 1 m ω (MeV) 1 1 783 1 1 m ρ (MeV) 1 1 763 1 1 κ 1 1 1.42 1 1 λ 1 1 0.0238 1 1 ζ 1 1 0.06 1 1 Λ v 1 1 0.03 1 1 Coupling 1 1 VF-RMF 1 1 FSUGold 1 1 g σN 1 1 14.57 1 1 10.59 1 1 g ωN 1 1 12.917 1 1 14.3 1 1 g ρN 1 1 10.29 1 1 11.7673 The effective nucleon mass calculated from Eq. (10) is shown by the solid line in Fig. 2 Fig. 2 . The decreasing of nucleon effective mass versus nucleon density is slower with the VF effects than that without, in agreement with the findings using other VF-RMF models [ 11 11 , 18 18 ]. The effective meson masses (on-shell and off-shell) calculated from Eqs. (11)–(17) are depicted in Fig. 3 Fig. 3 . Our results here agree with that of Ref. [ 18 18 ] as the same loop diagrams are involved. We note that the on-shell effective meson masses ( Fig. 3(a) Fig. 3(a) ) decrease at the normal density, which are indicated in most experiments and theoretical studies [ 11 11 - 18 18 11-18 ]. As noted in Ref. [ 18 18 ], the effective meson masses begin to increase at a high density, which are beyond the reach of current experiments. As the four-momentum transfer equals zero, the off-shell effective meson masses ( Fig. 3(b) Fig. 3(b) ) increase with the nucleon density, which is different from the on-shell ones. The EOS curve determined from Eqs. (21), (22) (when the neutron star matter reaches β equilibrium) is presented in Fig. 4 Fig. 4 together with the curve from the FSUGold model for a comparison. We see again that the EOS curve in our VF-RMF model is lower than that in the FSUGold model, similar to the findings in Ref. [ 18 18 ]. That is to say, the VF effects ’softened’ the EOS curve of the asymmetric neutron matter. In Fig. 5 Fig. 5 and Fig. 6 Fig. 6 , we presented the particle population for ( npeμ ) at different nucleon densities in the FSUGold model and VF-RMF model, respectively. The left panel is for Λ v =0.03 and the right panel for Λ v =0. We see that the population curves in Fig. 5 Fig. 5 and Fig. 6 Fig. 6 bear almost the same shapes, the differences are negligible. Now following standard procedures for solving the TOV equation with the help of EOS, we could obtain the mass (in units of solar mass, M ⊙ ) of the neutron star. The outcome is shown in Fig. 7 Fig. 7 and Table 3 Table 3 . The maximum mass of the ( npeμ ) neutron stars in our VF-RMF model is 1.35 M ⊙ , about 21% down in size compared to the FSUGold value, 1.71M_⊙[ 19 19 ], similar to that found in Ref. [ 18 18 ]. The curves obtained in other non-VF models[ 19 19 ] and the two recent astronomical observations, the binary-pulsar systems J 0348+0432 and J 1614-2230 [ 22 22 , 23 23 ], were also shown in Fig. 7 Fig. 7 for comparison. The implications will be briefly discussed in next section. We note in passing that a quark star with maximum mass over two solar mass was obtained in an improved quasiparticle model in Ref. [ 24 24 ]. 1 1 Parameter 1 1 VF-RMF 1 1 FSUGold 1 1 Λ ν =0.03 1 1 1.35 M ⊙ 1 1 1.71 M ⊙ 4 Conclusion 1 In summary, we considered the VF effects (from nucleon loop) in the framework of RMF that include additional isoscalar-isovector cross interaction terms for softening the symmetric energy of nuclear matter at a high density. For illustration, we considered in this work the simple case of determining the three nucleon-meson couplings ( g σN , g ωN , g ρN ) via some saturation properties, while keeping the rest of the parameters of the FSUGold model intact. Like studies using other RMF models, the VF effects tend to ’soften’ the EOS curve and other properties of the asymmetric neutron matter, and the maximum mass of such neutron stars is reduced from 1.71 M _⊙ to 1.35 M ⊙ . Given that the recent experimental/observational data of maximum neutron mass is about 2.0M ⊙ [ 22 22 , 23 23 ], it seems in principle to be a demanding job to explore how other and higher loop corrections within the RMF framework could affect the neutron properties before arriving at reliable theoretical conclusions. As we only determined the coupling constants in a very limited region of the whole parameter space, the results obtained here could not be used to infer too much. Further exploration of other regions of the parameter space for the couplings in Eq. (1) will be done in the near future. References N.K. Glendenning , Compact Stars . Berlin(Germany) : Springer-Verlag , 2000 . Compact Stars Z. Zhang , H. Shen , Relativistic equation of state for non-uniform nuclear matter . Nucl Tech , 2014 , 37 : 100514 . (in Chinese) DOI: 10.11889/j.0253-3219.2014.hjs.37.100514 http://doi.org/10.11889/j.0253-3219.2014.hjs.37.100514 Relativistic equation of state for non-uniform nuclear matter M. Yu , K.J. Duan , S.S. Wang , et al . A nuclear desity probe: isobaric yield ratio difference . Nucl Sci Tech , 2015 , 26 : S20503 . DOI: 10.13538/j.1001-8042/nst.26.S20503 http://doi.org/10.13538/j.1001-8042/nst.26.S20503 A nuclear desity probe: isobaric yield ratio difference X.Q. Liu , M.R. Huang , R. Wada , et al . Symmetry energy extraction from primary fragments in intermediate heavy-ion collisions . Nucl Sci Tech , 2015 , 26 : S20508 . DOI: 10.13538/j.1001-8042/nst.26.S20508 http://doi.org/10.13538/j.1001-8042/nst.26.S20508 Symmetry energy extraction from primary fragments in intermediate heavy-ion collisions J.D. Walecka , A theory of highly condensed matter . Ann Phys , 1974 , 83 : 491 - 529 . DOI: 10.1016/0003-4916(74)90208-5 http://doi.org/10.1016/0003-4916(74)90208-5 A theory of highly condensed matter B.D. Serot , J.D. Walecka , The relativistic nuclear many-body problem . In Negele J W and Vogt E , editors, Adv Nucl Phys , Plenum Press , 1986 , 16 : 1 . The relativistic nuclear many-body problem C.J. Horowitz , J. Piekarewicz , Neutron star structure and the neutron radius of 208Pb . Phys Rev Lett , 2001 , 86 : 5647 - 5650 . DOI: 10.1103/PhysRevLett.86.5647 http://doi.org/10.1103/PhysRevLett.86.5647 Neutron star structure and the neutron radius of 208Pb B.G. Todd-Rutel , J. Piekarewicz , Neutron-rich nuclei and neutron stars: A new accurately calibrated interaction for the study of neutron-rich matter . Phys Rev Lett , 2005 , 95 : 122501 . DOI: 10.1103/PhysRevLett.95.122501 http://doi.org/10.1103/PhysRevLett.95.122501 Neutron-rich nuclei and neutron stars: A new accurately calibrated interaction for the study of neutron-rich matter J. Piekarewicz , Pygmy dipole resonance as a constraint on the neutron skin of heavy nuclei . Phys Rev C , 2006 , 73 : 044325 . DOI: 10.1103/PhysRevC.73.044325 http://doi.org/10.1103/PhysRevC.73.044325 ; Pygmy dipole resonance as a constraint on the neutron skin of heavy nuclei Validating relativistic models of nuclear structure against theoretical, experimental, and observational constraints . Phys Rev C , 2007 , 76 : 064310 . DOI: 10.1103/PhysRevC.76.064310 http://doi.org/10.1103/PhysRevC.76.064310 Validating relativistic models of nuclear structure against theoretical, experimental, and observational constraints F.G. Fattoyev , J. Piekarewicz , Relativistic models of the neutron-star matter equation of state . Phys Rev C , 2010 , 82 : 025805 . DOI: 10.1103/PhysRevC.82.025805 http://doi.org/10.1103/PhysRevC.82.025805 Relativistic models of the neutron-star matter equation of state A. Bhattacharyya , S.K. Ghosh , S.C. Phatak , Effect of in-medium meson masses on nuclear matter properties . Phys Rev C , 1999 , 60 : 044903 . DOI: 10.1103/PhysRevC.60.044903 http://doi.org/10.1103/PhysRevC.60.044903 Effect of in-medium meson masses on nuclear matter properties G.E. Brown , M. Rho , Scaling effective Lagrangians in a dense medium . Phys Rev Lett , 1991 , 66 : 2720 - 2723 . DOI: 10.1103/PhysRevLett.66.2720 http://doi.org/10.1103/PhysRevLett.66.2720 Scaling effective Lagrangians in a dense medium H. Hatsuda , S.H. Lee , Phys Rev C , 1992 , 46 : R34(R) . DOI: 10.1103/PhysRevC.46.R34 http://doi.org/10.1103/PhysRevC.46.R34 Phys Rev C S. Sarkar , J. Alam , P. Roy , et al . Photons from hadronic matter at finite temperature . Nucl Phys A , 1998 , 634 : 206 - 230 . DOI: 10.1016/S0375-9474(98)00159-6 http://doi.org/10.1016/S0375-9474(98)00159-6 Photons from hadronic matter at finite temperature K. Ozawa , H. En’yo , H. Funahashi , et al . Observation of ρ/ω meson modification in nuclear matter . Phys Rev Lett , 2001 , 86 : 5019 - 5022 . DOI: 10.1103/PhysRevLett.86.5019 http://doi.org/10.1103/PhysRevLett.86.5019 Observation of ρ/ω meson modification in nuclear matter D. Trnka , G. Anton , J.C.S. Bacelar , et al . Observation of in-medium modifications of the ω meson . Phys Rev Lett , 2005 , 94 : 192303 . DOI: 10.1103/PhysRevLett.94.192303 http://doi.org/10.1103/PhysRevLett.94.192303 Observation of in-medium modifications of the ω meson B. Krusche , In-medium properties of hadrons . Int J Mod Phys A , 2007 , 22 : 406 - 415 . DOI: 10.1142/S0217751X07035598 http://doi.org/10.1142/S0217751X07035598 In-medium properties of hadrons X.H. Guo , B. Liu , M.H. Weng , Vacuum fluctuation effects on asymmetric nuclear matter . Phys Rev C , 2009 , 79 : 054316 . DOI: 10.1103/PhysRevC.79.054316 http://doi.org/10.1103/PhysRevC.79.054316 Vacuum fluctuation effects on asymmetric nuclear matter C. Wu , Z. Ren , Strange hadronic stars in relativistic mean-field theory with the FSUGold parameter set . Phys Rev C , 2011 , 83 : 025805 . DOI: 10.1103/PhysRevC.83.025805 http://doi.org/10.1103/PhysRevC.83.025805 Strange hadronic stars in relativistic mean-field theory with the FSUGold parameter set C. Wu , W.L. Qian , Y.G. Ma , et al . Neutron stars with kaon condensation in relativistic effective model . Int J Mod Phys E , 2013 , 22 : 1350026 . DOI: 10.1142/S0218301313500262 http://doi.org/10.1142/S0218301313500262 Neutron stars with kaon condensation in relativistic effective model H. Kurasawa , T. Suzuki , Effects of the Dirac sea on the Coulomb sum value for electron scattering . Nucl Phys A , 1988 , 490 : 571 - 584 . DOI: 10.1016/0375-9474(88)90014-0 http://doi.org/10.1016/0375-9474(88)90014-0 Effects of the Dirac sea on the Coulomb sum value for electron scattering J. Antoniadis , P.C.C. Freire , N. Wex , et al . A massive pulsar in compact relativistic binary . Science , 2013 , 340 : 1233232 . DOI: 10.1126/science.1233232 http://doi.org/10.1126/science.1233232 A massive pulsar in compact relativistic binary P.B. Demorest , T. Pennucci , S.M. Ransom , et al . A two-solar-mass neutron star measured using Shapiro delay . Nature , 2010 , 467 : 1081 - 1083 . DOI: 10.1038/nature09466 http://doi.org/10.1038/nature09466 A two-solar-mass neutron star measured using Shapiro delay T. Zhao , Y. Yan , X.L. Luo , et al . Study of rotational quark stars and hybrid stars based on the latest equation of state and observation data . Phys Rev D , 2015 , 91 : 034018 . DOI: 10.1103/PhysRevD.91.034018 http://doi.org/10.1103/PhysRevD.91.034018 Study of rotational quark stars and hybrid stars based on the latest equation of state and observation data 10.1007/s41365-016-0023-8 Supported by National Natural Science Foundation of China (No. 11435005) and the Major State Basic Research Development Program in China (No. 2014CB845402) 2016-01-13 2016-02-01 2016-02-05 2016-03-02 2016-02-20 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/s41365-016-0023-8 http://dx.doi.org/10.1007/s41365-016-0023-8 . 2016 Nuclear Science and Techniques 01 27