Flavor-mixing induced by the mismatched vector interactions at finite μ_I

Flavor-mixing induced by the mismatched vector interactions at finite μ I 1 corresp first-author Corresponding author, zhaozhang@pku.org.cn Corresponding author, zhaozhang@pku.org.cn zhaozhang@pku.org.cn 1 School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China The relation between the vector-isoscalar and vector-isovector interactions in the two-flavor Nambu-Jona-Lasinio (NJL) model is investigated under the different constraints from QCD. We demonstrate that the flavor-mixing can be induced by the mismatched vector-isoscalar and vector-isovector interactions at finite baryon chemical potential μ and isospin chemical potential μ I . The effect of this non-anomaly flavor-mixing on the possible separate chiral transitions at nonzero μ I is studied under the assumption of the effective restoration of U (1) A symmetry. We find that for the weak isospin asymmetry, the two separate phase boundaries found previously can be converted into one only if the vector-isovector coupling <math id="M1"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> is significantly stronger than the vector-isoscalar one <math id="M2"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> without the axial anomaly. Chiral phase transition QCD critical point Flavor-mixing Isospin chemical potential I. INTRODUCTION 1 Plotting the map of QCD phase diagram at finite temperature T and quark chemical potential μ has attracted growing interests. Especially, a chiral critical endpoint is predicted by some model studies, which separates the crossover and the first order phase transition. Currently, the lattice QCD computations at finite density still struggle in a limited range of μ [ 1 1 ]. So the existence and location of the critical endpoint are still under debate due to the lack of reliable theoretical methods for the non-perturbative dense QCD. Theoretically, fluctuations of conserved charges, such as net-baryon, net-charge and net-strangeness, are predicted to be sensitive to the correlation length of the system [ 2 2 - 4 4 2-4 ] and directly connected to some susceptibilities [ 5 5 , 6 6 ]. Thus, the experimental data related to these quantities can serve as powerful tools to probe the critical endpoint in heavy-ion collisions: the search for such a point is ongoing at RHIC (BES) [ 7 7 - 9 9 7-9 ] and will be performed in the future facilities in GSI (FAIR) and JINR (NICA). Moreover, unconventional multiple chiral critical endpoints are also proposed: it is found in [ 10 10 , 11 11 ] that the finite isospin chemical potential μ I may lead to two critical endpoints; when considering the color superconductivity (CSC), the low-temperature critical endpoint(s) may appear due to the interplay between the chiral and diquark condensates [ 12 12 - 17 17 12-17 ]. Besides T and μ , the U (1) A anomaly may impact the QCD phase transition significantly [ 18 18 ]. This point has been confirmed in model studies or Ginzberg-Landau analyses, where the U (1) A anomaly is usually incorporated by introducing the Kabayashi-Maskawa-’t Hooft (KMT) interaction [ 19 19 - 21 21 19-21 ]. The KMT interaction explicitly breaks the U (1) A symmetry and gives rise to the flavor-mixing among light quarks: the dynamical mass of u quark may contain contributions from both d and s quark condensates; the diquark condensate for u-d pairing at moderate or high baryon density may make contribution to the s quark mass. Consequently, the U (1) A anomaly may affect not only the properties of the traditional critical endpoint [ 22 22 ], but also that of the unconventional one: the separation of chiral transition due to finite μ I [ 10 10 , 11 11 ] may be removed by the anomaly flavor-mixing [ 23 23 ]; a new critical endpoint at low temperature could be induced in the presence of the color flavor locking (CFL) CSC [ 13 13 ]. However, the recent lattice calculations indicate that the U (1) A symmetry may be restored obviously near and above T c for zero μ [ 24 24 , 25 25 ]. The effective restoration of the U (1) A symmetry would influence the universality class and critical properties of the chiral transition [ 26 26 ]. Phenomenologically, the model studies suggest that the properties of the conventional critical endpoint are quite sensitive to the degree of U (1) A restoration [ 22 22 ]. Moreover, if the anomaly related flavor-mixing is very weak near the phase boundary, the two critical endpoints due to the isospin asymmetry [ 10 10 , 11 11 ] could still be possible due to the decouple of light quarks. On the other hand, it is also probable that the non-anomaly flavor-mixing can be induced by other ingredients of QCD, especially under some condition. The main purpose of this work is to study the possible non-anomaly flavor-mixing of light quarks and its effect on the chiral phase transition under the isospin asymmetry. In particular, we will focus on the fate of the two critical endpoints due to the separate chiral transitions found in [ 10 10 , 11 11 ] with the assumption of the effective restoration of U (1) A symmetry near the phase boundary. Our starting point is the four-quark vector interactions with different coupling strengths in the isovector and isoscalar channels. The effect of vector interactions on the chiral transition has been extensively studied in the NJL-type model of QCD. A well-known result is that the chiral transition at finite μ is weakened by the vector-isoscalar interaction <math id="M3"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msub><mi>γ</mi><mi>μ</mi></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math> and the critical point disappears for strong <math id="M4"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> [ 12 12 , 15 15 , 22 22 , 27 27 ]. It is also found that the <math id="M5"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> in a proper range can lead to the new low-temperature critical endpoint(s) [ 12 12 ] when considering the two-flavor CSC, especially with the constraint of electric-charge neutrality [ 15 15 , 17 17 ]. In general, <math id="M6"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> and the isovector coupling <math id="M7"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> are independent coupling constants under the chiral symmetry. In the literature, the non-anomaly flavor-mixing due to the mismatched vector interactions in the vacuum has been discussed in Ref. [ 28 28 ] based on a three-flavor NJL model. We extend such a work to finite T and μ to study the effect of the non-anomaly flavor-mixing on the chiral transition with isospin asymmetry. II. EXTENDED NJL-TYPE MODEL WITH MISMATCHED VECTOR INTERACTIONS 1 A. The general four-quark interaction model with mismatched vector interactions under the chiral symmetry 2 We start with the following Lagrangian of four-quark interaction model for two-flavor QCD <math display="block" id="M8"><mrow><msup><mi mathvariant="script">L</mi><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><msubsup><mi mathvariant="script">L</mi><mrow><mtext>sym</mtext></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><msubsup><mi mathvariant="script">L</mi><mrow><mtext>det</mtext></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup><mn>,</mn></mrow></math> with <math display="block" id="M9"><mtable columnalign="left"><mtr><mtd><msubsup><mi mathvariant="script">L</mi><mrow><mtext>sym</mtext></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow><mn>3</mn></munderover></mstyle><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>g</mi><mrow><mi>v</mi><mn>2</mn></mrow></msub><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow><mn>3</mn></munderover></mstyle><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><msub><mi>γ</mi><mi>μ</mi></msub><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><msub><mi>γ</mi><mi>μ</mi></msub><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>g</mi><mrow><mi>v</mi><mn>3</mn></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mn>0</mn></msup><msub><mi>γ</mi><mi>μ</mi></msub><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mn>0</mn></msup><msub><mi>γ</mi><mi>μ</mi></msub><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>g</mi><mrow><mi>v</mi><mn>4</mn></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mn>0</mn></msup><msub><mi>γ</mi><mi>μ</mi></msub><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mn>0</mn></msup><msub><mi>γ</mi><mi>μ</mi></msub><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo><mn>,</mn></mtd></mtr></mtable></math> and <math display="block" id="M10"><mtable columnalign="left"><mtr><mtd><msubsup><mi mathvariant="script">L</mi><mrow><mtext>det</mtext></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub><mo>{</mo><mi>d</mi><mi>e</mi><mi>t</mi><mo stretchy="false">[</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>γ</mi><mn>5</mn></msub><mo stretchy="false">)</mo><mi>ψ</mi><mo stretchy="false">]</mo><mo>+</mo><mi>h</mi><mn>.</mn><mi>c</mi><mn>.</mn><mo>}</mo></mtd></mtr><mtr><mtd><mo>=</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mover accent="true"><mi>τ</mi><mo>→</mo></mover><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mover accent="true"><mi>τ</mi><mo>→</mo></mover><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo><mn>,</mn></mtd></mtr></mtable></math> where τ 0 , and <math id="M11"><mover accent="true"><mi>τ</mi><mo>→</mo></mover></math> refer to the unit matrix and Pauli matrices in the flavor space, respectively. The former term <math id="M12"><mrow><msubsup><mi mathvariant="script">L</mi><mrow><mtext>sym</mtext></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> in (1) is the general Fierz-invariant form of the four-quark interactions in color-singlet channels which respecting the global flavor symmetries of SU (2) V ⊗ SU (2) A ⊗ U (1) V ⊗ U (1) A [ 29 29 ]. The latter one <math id="M13"><mrow><msubsup><mi mathvariant="script">L</mi><mrow><mtext>det</mtext></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> is the KMT interaction induced by the gauge configurations of instanton and anti-instanton [ 19 19 , 20 20 ], which only possesses the SU (2) V ⊗ SU (2) A ⊗ U (1) V global flavor symmetries. As mentioned, we will focus on the flavor-mixing arising from the mismatched vector interactions at finite density. We see that three of four independent coupling constants in <math id="M14"><mrow><msubsup><mi mathvariant="script">L</mi><mrow><mtext>sym</mtext></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup></mrow></math> are related to the vector and axial vector interactions. Generally, the nozero sum g v 3 + g v 4 implies that the vector coupling strength in the isovector channel is different from that in the isoscalar one. Similarly, the non-vanishing g v 3 - g v 4 indicates the axial-vector interactions are also mismatched in the isovector and isoscalar channels. How the vector coupling difference gives rise to the non-anomaly flavor-mixing at finite density will be detailed in next section. The axial-vector interaction may be responsible for the deviation of the chiral magnetic effect in the recent lattice calculation compared to the analytic formula, as proposed in Ref. [ 30 30 ]. Here we mainly study the chiral phase transition in the MFA, the axial-vector interactions in Lagrangian (1) will be ignored. Thus we only consider the the following effective Lagrangian, <math display="block" id="M15"><mtable columnalign="left"><mtr><mtd><msubsup><mi mathvariant="script">L</mi><mrow><mtext>eff</mtext></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow><mn>3</mn></munderover></mstyle><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mover accent="true"><mi>τ</mi><mo>→</mo></mover><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mover accent="true"><mi>τ</mi><mo>→</mo></mover><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msub><mi>γ</mi><mi>μ</mi></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mover accent="true"><mi>τ</mi><mo>→</mo></mover><msub><mi>γ</mi><mi>μ</mi></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mn>,</mn></mtd></mtr></mtable></math> where the independent coupling constants are reduced to four. B. Unequal vector coupling constants in the mean field Hartree-Fork approximation 2 Here we stress that the vector coupling difference in the MFA can also arise from a very popular version of the NJL model [ 29 29 ], <math display="block" id="M16"><mtable columnalign="left"><mtr><mtd><msup><mi mathvariant="script">L</mi><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow><mn>3</mn></munderover></mstyle><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>+</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mover accent="true"><mi>τ</mi><mo>→</mo></mover><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mover accent="true"><mi>τ</mi><mo>→</mo></mover><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>g</mi><mi>v</mi></msub><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow><mn>3</mn></munderover></mstyle><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><msub><mi>γ</mi><mi>μ</mi></msub><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><msub><mi>γ</mi><mi>μ</mi></msub><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo><mn>,</mn></mtd></mtr></mtable></math> in which only one vector coupling g v is adopted. In the Hartree approximation, there is no difference between the coupling strengths of the two vector interactions at the mean field level for Lagrangian (5). However, the effective vector couplings (in the sense of direct interaction) in the isoscalar and isovector channels will differ from each other if the Fock contribution is also considered. For a four-fermion interaction, the Fock contribution can be easily evaluated according to its Fierz transformation [ 29 29 ]. Taking into account the exchange terms, the effective direct four-quark interactions of the Lagrangian (5) take the following form <math display="block" id="M17"><mtable columnalign="left"><mtr><mtd><msup><mi mathvariant="script">L</mi><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><msub><mrow/><mrow><mtext>eff</mtext><mo>−</mo><mtext>direct</mtext></mrow></msub><mo>=</mo><msup><mi mathvariant="script">L</mi><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msup><mo>+</mo><msubsup><mi mathvariant="script">L</mi><mrow><mtext>Fock</mtext></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup></mtd></mtr><mtr><mtd><mo>=</mo><mo stretchy="false">(</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mover accent="true"><mi>τ</mi><mo>→</mo></mover><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>+</mo><mo stretchy="false">(</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo>−</mo><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub><mo>−</mo><mfrac><mrow><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub></mrow><mrow><mn>2</mn><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mover accent="true"><mi>τ</mi><mo>→</mo></mover><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mi>i</mi><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><msub><mi>g</mi><mi>v</mi></msub><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>a</mi><mo>=</mo><mn>0</mn></mrow><mn>3</mn></munderover></mstyle><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><msub><mi>γ</mi><mi>μ</mi></msub><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mi>a</mi></msup><msub><mi>γ</mi><mi>μ</mi></msub><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>g</mi><mi>v</mi></msub></mrow><mrow><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub></mrow><mrow><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mn>0</mn></msup><msub><mi>γ</mi><mi>μ</mi></msub><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mtd></mtr><mtr><mtd><mo>−</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mi>g</mi><mi>v</mi></msub></mrow><mrow><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfrac><mrow><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub></mrow><mrow><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msup><mi>τ</mi><mn>0</mn></msup><msub><mi>γ</mi><mi>μ</mi></msub><msub><mi>γ</mi><mn>5</mn></msub><mi>ψ</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mn>,</mn></mtd></mtr></mtable></math> where N c is the color number of the quarks. The effective Lagrangian (6) clearly shows that the exchange terms give rise to the vector coupling difference in the Hartree-Fock approximation (HFA), which is at the order of O (1/ N c ) compared to g s 1 and g v . Note that the similar result in a three-flavor NJL model has been given in [ 28 28 ]. If both g v and g s 1 in (5) originate from the color current-current interaction <math id="M18"><mrow><mi>g</mi><msup><mrow><mo stretchy="false">(</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><msub><mi>γ</mi><mi>μ</mi></msub><msubsup><mi>λ</mi><mi>c</mi><mi>a</mi></msubsup><mi>ψ</mi><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></math> , they fulfill the relation g v = g s 1 /2 according to the Fierz transformation. In this case, the vector coupling difference shown in (6) becomes <math display="block" id="M19"><mrow><mi>δ</mi><msub><mi>g</mi><mi>v</mi></msub><mo>=</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>−</mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>=</mo><mfrac><mn>2</mn><mrow><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><msub><mi>g</mi><mi>v</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub></mrow><mrow><msub><mi>N</mi><mi>c</mi></msub></mrow></mfrac><mn>.</mn></mrow></math> This equation indicates that <math id="M20"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> in (4) may be larger than <math id="M21"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> and their difference is considerable compared to g v or g s 1 for N c =3. C. Constraints on the vector interactions from the lattice chiral curvatures 2 Even Eq. (7) implies that the coupling <math id="M22"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> is weaker than <math id="M23"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> , it is also possible that <math id="M24"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> may be stronger than <math id="M25"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> . This can be understood from the curvature difference for the chiral phase transition at finite baryon and isospin chemical potentials obtained in recent lattice calculations [ 31 31 ]. For small baryon and isospin densities, the chemical potential dependence of the pseudo-critical temperature for the chiral crossover can be expressed as <math display="block" id="M26"><mrow><msub><mi>T</mi><mi>c</mi></msub><mo stretchy="false">(</mo><msub><mi>μ</mi><mi>q</mi></msub><mn>,</mn><msub><mi>μ</mi><mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>T</mi><mi>c</mi></msub><mo>+</mo><msub><mi>A</mi><mi>q</mi></msub><msubsup><mi>μ</mi><mi>q</mi><mn>2</mn></msubsup><mo>+</mo><msub><mi>B</mi><mi>i</mi></msub><msubsup><mi>μ</mi><mi>i</mi><mn>2</mn></msubsup><mo>+</mo><mi>O</mi><mo stretchy="false">(</mo><msubsup><mi>μ</mi><mrow><mi>q</mi><mo>/</mo><mi>i</mi></mrow><mn>4</mn></msubsup><mn>,</mn><msubsup><mi>μ</mi><mi>q</mi><mn>2</mn></msubsup><msubsup><mi>μ</mi><mi>i</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><mtext> </mtext><mn>,</mn></mrow></math> where T c is the chiral pseudo-critical temperature at zero quark chemical potential (In this subsection, μ q and μ i are used to refer to the quark baryon and isospin chemical potentials, respectively). Notice that T c ( μ q , μ i ) is an even function of μ q/i [ 32 32 ]. So at the order of <math id="M27"><mrow><msubsup><mi>μ</mi><mrow><mi>q</mi><mo>/</mo><mi>i</mi></mrow><mn>2</mn></msubsup></mrow></math> , we can expand <math id="M28"><mrow><msub><mi>T</mi><mi>c</mi></msub><mo stretchy="false">(</mo><msubsup><mi>μ</mi><mrow><mi>q</mi><mo>/</mo><mi>i</mi></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo></mrow></math> as <math display="block" id="M29"><mrow><msub><mi>T</mi><mi>c</mi></msub><mo stretchy="false">(</mo><msubsup><mi>μ</mi><mrow><mi>q</mi><mo>/</mo><mi>i</mi></mrow><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>=</mo><msub><mi>T</mi><mi>c</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msub><mi>κ</mi><mrow><mi>q</mi><mo>/</mo><mi>i</mi></mrow></msub><mfrac><mrow><msubsup><mi>μ</mi><mrow><mi>q</mi><mo>/</mo><mi>i</mi></mrow><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>T</mi><mi>c</mi><mn>2</mn></msubsup></mrow></mfrac><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where the two chiral curvatures are defined as <math display="block" id="M30"><mrow><msub><mi>κ</mi><mrow><mi>q</mi><mo>/</mo><mi>i</mi></mrow></msub><mo>=</mo><mo>−</mo><msub><mi>T</mi><mi>c</mi></msub><mfrac><mrow><mi>d</mi><msub><mi>T</mi><mi>c</mi></msub><mo stretchy="false">(</mo><msup><mi>μ</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><mrow><mi>d</mi><msubsup><mi>μ</mi><mrow><mi>q</mi><mo>/</mo><mi>i</mi></mrow><mn>2</mn></msubsup></mrow></mfrac><msub><mo>|</mo><mrow><mi>μ</mi><mo>=</mo><mn>0</mn></mrow></msub><mn>.</mn></mrow></math> The lattice QCD simulation in [ 31 31 ] suggests that the curvature κ q is about 10% greater than k i . Recently, the role of <math id="M31"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> on the determination of κ q has been studied in a Polyakov-loop enhanced three-flavor NJL model [ 22 22 ]. It is found that κ q decreases with <math id="M32"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> and to reproduce the lattice κ q the <math id="M33"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> must keep relatively larger value compared to g s . The authors of Ref. [ 22 22 ] then propose the lattice κ q can be used as a useful constraint on <math id="M34"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> . We can directly extend this idea to determine κ i by replacing μ q with μ i . As will be demonstrated in the next section, the coupling <math id="M35"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> influences the curvature κ i in the similar way as <math id="M36"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> does on κ q . In particular, κ i and κ q obtained at the MFA of the two-flavor NJL model will take the same value for <math id="M37"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>=</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> . In other words, the lattice curvature difference between κ i and κ q can be regarded as an useful evidence for the unequal vector coupling strengths. Since the two-flavor lattice calculation in [ 31 31 ] indicates that κ i is less than κ q , we thus infer that <math id="M38"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> may be larger than <math id="M39"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> near the chiral phase boundary for zero and small quark chemical potential. Following the spirit of Ref. [ 22 22 ], our numerical study suggests that <math id="M40"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> is about 10% larger than <math id="M41"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> near T c according to the lattice curvatures in [ 31 31 ]. Note that this conclusion is quite different from the estimation given in (7). D. Constraints on the vector interactions from the couplings of vector mesons to nucleons and lattice susceptibilities 2 In Ref. [ 33 33 ], it is argued that the ratio of the couplings of ω and ρ mesons to nucleons can be used as a constraint on the vector coupling difference. In the chirally broken phase, the empirical value for this ratio is given by g ωNN / g ρ NN 3, whereas in the chirally symmetric phase it is expected to be one. It is then proposed that the ratio <math id="M42"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>/</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> is located in the range from 1/3 to 1. In addition, another quite similar estimation is given in Ref. [ 34 34 ], where the vector coupling difference is expressed as the function of two susceptibilities χ q and χ I under some assumptions. Using the lattice data for these susceptibilities as input, it is found that <math id="M43"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> is always less than <math id="M44"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> : their difference is quite large below T c which approaches zero rapidly above T c for zero chemical potential. All the arguments given in the above subsections suggest that the vector interactions are repulsive (namely, <math id="M45"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> and <math id="M46"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> are all positive), but the relation between the <math id="M47"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> and <math id="M48"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> remains uncertain. In the following study, <math id="M49"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> and <math id="M50"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> in (4) will be treated as the free parameters due to these uncertainties. III. VECTOR-INTERACTION INDUCED FLAVOR-MIXING AND THE THERMAL DYNAMICAL POTENTIAL AT FINITE BARYON AND ISOSPIN CHEMICAL POTENTIALS 1 In this section, we shall demonstrate that the vector coupling difference can lead to non-anomaly flavor-mixing at finite baryon and isospin densities. The full Lagrangian of two-flavor NJL model with the interaction (4) reads <math display="block" id="M51"><mrow><mi mathvariant="script">L</mi><mo>=</mo><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mrow><mo>(</mo><mrow><mi>i</mi><msub><mo>∂</mo><mi>μ</mi></msub><msup><mi>γ</mi><mi>μ</mi></msup><mo>+</mo><msub><mi>γ</mi><mn>0</mn></msub><mover accent="true"><mi>μ</mi><mo>^</mo></mover><mo>−</mo><msub><mover accent="true"><mi>m</mi><mo>^</mo></mover><mn>0</mn></msub></mrow><mo>)</mo></mrow><mi>ψ</mi><mo>+</mo><msubsup><mi mathvariant="script">L</mi><mrow><mtext>eff</mtext></mrow><mrow><mo stretchy="false">(</mo><mn>4</mn><mo stretchy="false">)</mo></mrow></msubsup><mn>,</mn></mrow></math> where the quark chemical potentials are introduced and <math id="M52"><mrow><msub><mover accent="true"><mi>m</mi><mo>^</mo></mover><mn>0</mn></msub></mrow></math> = diag( m u , m d ) is the current quark matrix. We shall adopt the isospin symmetric quark masses with m u = m d m 0 . The <math id="M53"><mover accent="true"><mi>μ</mi><mo>^</mo></mover></math> in the Lagrangian (11) is the matrix of the quark chemical potentials which takes form <math display="block" id="M54"><mrow><mover accent="true"><mi>μ</mi><mo>^</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mtable><mtr><mtd><mrow><msub><mi>μ</mi><mi>u</mi></msub></mrow></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mrow/></mtd><mtd><mrow><msub><mi>μ</mi><mi>d</mi></msub></mrow></mtd></mtr></mtable><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mtable><mtr><mtd><mrow><mi>μ</mi><mo>−</mo><msub><mi>μ</mi><mi>I</mi></msub></mrow></mtd><mtd><mrow/></mtd></mtr><mtr><mtd><mrow/></mtd><mtd><mrow><mi>μ</mi><mo>+</mo><msub><mi>μ</mi><mi>I</mi></msub></mrow></mtd></mtr></mtable><mo stretchy="false">)</mo><mn>,</mn></mrow></math> with <math display="block" id="M55"><mrow><mi>μ</mi><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mi>u</mi></msub><mo>+</mo><msub><mi>μ</mi><mi>d</mi></msub></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mi>B</mi></msub></mrow><mn>3</mn></mfrac><mi/><mi/><mi/><mtext>and</mtext><mi/><mi/><mi/><msub><mi>μ</mi><mi>I</mi></msub><mo>=</mo><mfrac><mrow><msub><mi>μ</mi><mi>u</mi></msub><mo>−</mo><msub><mi>μ</mi><mi>d</mi></msub></mrow><mn>2</mn></mfrac><mo>=</mo><mfrac><mrow><mi>δ</mi><mi>μ</mi></mrow><mn>2</mn></mfrac><mn>.</mn></mrow></math> In (13), μ B ( μ I ) is the baryon (isospin) chemical potential. Note that the definition of the isospin chemical potential in the quark level is different from that in the nucleon level. For more details on the role of isospin symmetry energy in nuclear matter, the reader can refer to [ 35 35 - 38 38 35-38 ] and references therein. At finite densities, the quark chemical potentials are shifted by the vector interactions. Here we use μ’ to denote the modified quark chemical potential. Note that the u quark density is different from the d quark one under the isospin asymmetry. The shifted quark chemical potentials take the form <math display="block" id="M56"><mtable columnalign="left"><mtr><mtd><msub><msup><mi>μ</mi><mo>′</mo></msup><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><msub><mi>μ</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>−</mo><mn>2</mn><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mi>u</mi></msub><mo>+</mo><msub><mi>ρ</mi><mi>d</mi></msub><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>−</mo><msub><mi>ρ</mi><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo>=</mo><msub><mi>μ</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>−</mo><mn>2</mn><mo stretchy="false">(</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>+</mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo stretchy="false">)</mo><msub><mi>ρ</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>−</mo><mn>2</mn><mo stretchy="false">(</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>−</mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo stretchy="false">)</mo><msub><mi>ρ</mi><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></msub><mn>,</mn></mtd></mtr></mtable></math> or <math display="block" id="M57"><mrow><msup><mi>μ</mi><mo>′</mo></msup><mo>=</mo><mi>μ</mi><mo>−</mo><mn>2</mn><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mi>u</mi></msub><mo>+</mo><msub><mi>ρ</mi><mi>d</mi></msub><mo stretchy="false">)</mo><mn>,</mn><mtext> </mtext><msub><mi>μ</mi><mi>I</mi></msub><mo>'</mo><mo>=</mo><msub><mi>μ</mi><mi>I</mi></msub><mo>−</mo><mn>2</mn><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mi>u</mi></msub><mo>−</mo><msub><mi>ρ</mi><mi>d</mi></msub><mo stretchy="false">)</mo></mrow></math> where <math display="block" id="M58"><mrow><msub><mi>ρ</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mo>〈</mo><msubsup><mi>ψ</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><mo>†</mo></msubsup><msub><mi>ψ</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>〉</mo><mn>,</mn></mrow></math> is the u (d) quark number density. Eq. (14) clearly shows that due to the vector coupling difference, not only ρ u but also ρ d give contribution to the effective chemical potential of u quark, and vise versa. This implies that the flavor-mixing arises due to the vector interaction. As mentioned, this mixing has nothing to do with the axial anomaly. The modified chemical potentials can also be rearranged as Eq. (15), which indicates that μ and μ I are shifted by the isoscalar and isovector vector interactions, respectively. Formally, the non-anomaly flavor-mixing shown in (14) for the modified chemical potentials is quite similar to the anomaly flavor-mixing for the constituent quark masses induced by the instantons, namely: <math display="block" id="M59"><mrow><msub><mi>M</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><msub><mi>m</mi><mn>0</mn></msub><mo>−</mo><mn>4</mn><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><msub><mi>ϕ</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>−</mo><mn>4</mn><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub><msub><mi>ϕ</mi><mrow><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mo stretchy="false">)</mo></mrow></msub><mn>,</mn></mrow></math> where <math display="block" id="M60"><mrow><msub><mi>ϕ</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mo>〈</mo><msub><mover accent="true"><mi>ψ</mi><mo>¯</mo></mover><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><msub><mi>ψ</mi><mrow><mi>u</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo>〉</mo><mn>,</mn></mrow></math> is the u (d) quark condensate. Using the conventional technique, the mean field thermal dynamical potential of the Lagrangian (11) is expressed as <math display="block" id="M61"><mtable columnalign="left"><mtr><mtd><mi>Ω</mi><mo stretchy="false">(</mo><mi>T</mi><mn>,</mn><msub><mi>μ</mi><mi>u</mi></msub><mn>,</mn><msub><mi>μ</mi><mi>d</mi></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mstyle displaystyle="true"><munder><mo>∑</mo><mrow><mi>f</mi><mo>=</mo><mi>u</mi><mn>,</mn><mi>d</mi></mrow></munder></mstyle><msub><mi>Ω</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mn>,</mn><msub><msup><mi>μ</mi><mo>′</mo></msup><mi>f</mi></msub><mo>;</mo><msub><mi>M</mi><mi>f</mi></msub><mo stretchy="false">)</mo><mo>+</mo><mn>2</mn><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><msubsup><mi>ϕ</mi><mi>u</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>ϕ</mi><mi>d</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>+</mo><mn>4</mn><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub><msub><mi>ϕ</mi><mi>u</mi></msub><msub><mi>ϕ</mi><mi>d</mi></msub></mtd></mtr><mtr><mtd><mo>−</mo><mo stretchy="false">(</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>+</mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msubsup><mi>ρ</mi><mi>u</mi><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>ρ</mi><mi>d</mi><mn>2</mn></msubsup><mo stretchy="false">)</mo><mo>−</mo><mn>2</mn><mo stretchy="false">(</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>−</mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo stretchy="false">)</mo><msub><mi>ρ</mi><mi>u</mi></msub><msub><mi>ρ</mi><mi>d</mi></msub><mn>,</mn></mtd></mtr></mtable></math> where <math id="M62"><mrow><msub><mi>Ω</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mn>,</mn><msub><msup><mi>μ</mi><mo>′</mo></msup><mi>f</mi></msub><mo>;</mo><msub><mi>M</mi><mi>f</mi></msub><mo stretchy="false">)</mo></mrow></math> is the contribution of a quasi-particle gas of the flavor f which takes the form <math display="block" id="M63"><mtable columnalign="left"><mtr><mtd><msub><mi>Ω</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>T</mi><mn>,</mn><msub><msup><mi>μ</mi><mo>′</mo></msup><mi>f</mi></msub><mo>;</mo><msub><mi>M</mi><mi>f</mi></msub><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo>−</mo><mn>2</mn><msub><mi>N</mi><mi>c</mi></msub><mi>T</mi><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mi>p</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><mo stretchy="false">[</mo><mi>ln</mi><mo stretchy="false">[</mo><mn>1</mn><mo>+</mo><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mo stretchy="false">(</mo><msub><mi>E</mi><mi>f</mi></msub><mo>−</mo><msub><msup><mi>μ</mi><mo>′</mo></msup><mi>f</mi></msub><mo stretchy="false">)</mo><mo>/</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>+</mo><mi>ln</mi><mo stretchy="false">[</mo><mn>1</mn><mo>+</mo><mi>exp</mi><mo stretchy="false">(</mo><mo>−</mo><mo stretchy="false">(</mo><msub><mi>E</mi><mi>f</mi></msub><mo>+</mo><msub><msup><mi>μ</mi><mo>′</mo></msup><mi>f</mi></msub><mo stretchy="false">)</mo><mo>/</mo><mi>T</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mtd></mtr><mtr><mtd><mo>−</mo><mn>2</mn><msub><mi>N</mi><mi>c</mi></msub><mstyle displaystyle="true"><mrow><mo>∫</mo></mrow></mstyle><mfrac><mrow><msup><mi>d</mi><mn>3</mn></msup><mi>p</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><msub><mi>E</mi><mi>f</mi></msub><mi>θ</mi><mo stretchy="false">(</mo><msup><mi>Λ</mi><mn>2</mn></msup><mo>−</mo><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>2</mn></msup><mo stretchy="false">)</mo><mn>,</mn></mtd></mtr></mtable></math> with the quasi-particle energy <math id="M64"><mrow><msub><mi>E</mi><mi>f</mi></msub><mo>=</mo><msqrt><mrow><msup><mover accent="true"><mi>p</mi><mo>→</mo></mover><mn>2</mn></msup><mo>+</mo><msubsup><mi>M</mi><mi>f</mi><mn>2</mn></msubsup></mrow></msqrt></mrow></math> . The Λ in Eq. (20) is the parameter of three-momentum cutoff in the NJL model. We see that besides the modified chemical potential <math id="M65"><mrow><msub><mi>μ</mi><msup><mi>f</mi><mo>′</mo></msup></msub></mrow></math> , the flavor-mixing due to the vector coupling difference is also explicitly demonstrated in Eq. (19) via the direct coupling between ρ u and ρ d . Minimizing the thermal dynamical potential Eq. (19), the motion equations for the mean fields ϕ u , ϕ d , ρ u and ρ d are determined through the coupled equations <math display="block" id="M66"><mrow><mfrac><mrow><mo>∂</mo><mi>Ω</mi></mrow><mrow><mo>∂</mo><msub><mi>ϕ</mi><mi>u</mi></msub></mrow></mfrac><mo>=</mo><mn>0,</mn><mtext> </mtext><mfrac><mrow><mo>∂</mo><mi>Ω</mi></mrow><mrow><mo>∂</mo><msub><mi>ϕ</mi><mi>d</mi></msub></mrow></mfrac><mo>=</mo><mn>0,</mn><mtext> </mtext><mfrac><mrow><mo>∂</mo><mi>Ω</mi></mrow><mrow><mo>∂</mo><msub><mi>ρ</mi><mi>u</mi></msub></mrow></mfrac><mo>=</mo><mn>0,</mn><mtext> </mtext><mfrac><mrow><mo>∂</mo><mi>Ω</mi></mrow><mrow><mo>∂</mo><msub><mi>ρ</mi><mi>d</mi></msub></mrow></mfrac><mo>=</mo><mn>0.</mn></mrow></math> This set of equations is then solved for the fields ϕ u , ϕ d , ρ u and ρ d as functions of the temperature and chemical potentials. IV. FATE OF THE SEPARATE CHIRAL TRANSITIONS WITH NON-ANOMALY FLAVOR-MIXING 1 As mentioned, the separate chiral transitions due to finite μ I [ 10 10 , 11 11 ] can be removed by the flavor-mixing induced by the axial anomaly [ 23 23 ]. Since the instanton density may be suppressed significantly near the phase boundary, we revisit this problem by taking into account the non-anomaly flavor-mixing due to the mismatched vector interactions. For comparison, we follow the notations in Ref. [ 23 23 ] and introduce two parameters α and g s which are defined as <math display="block" id="M67"><mrow><msub><mi>g</mi><mrow><mi>s</mi><mn>1</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo stretchy="false">)</mo><msub><mi>g</mi><mi>s</mi></msub><mn>,</mn><mtext> </mtext><mtext> </mtext><mtext> </mtext><msub><mi>g</mi><mrow><mi>s</mi><mn>2</mn></mrow></msub><mo>=</mo><mi>α</mi><msub><mi>g</mi><mi>s</mi></msub><mn>,</mn></mrow></math> here α means the ratio of the KMT interaction in the scalar-pseudoscalar channel, which is treated as a free parameter in the following calculations. The other model parameters, namely the current quark mass m 0 , the scalar coupling constant g s and the three-momentum cutoff Λ are all adopted from [ 23 23 ]. A. Fate of separate chiral transitions under the weak isospin asymmetry without the axial anomaly 2 The role of the mismatched vector interactions on the separation of chiral transition at finite T - μ under the weak isospin asymmetry is investigated by switching off the KMT interaction. We focus on whether the two critical endpoints found previously could be ruled out by the non-anomaly flavor-mixing without the help of the axial anomaly. We first study the cases for <math id="M68"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>></mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> with a fixed small coupling <math id="M69"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>=</mo><mn>0.2</mn><msub><mi>g</mi><mi>s</mi></msub></mrow></math> under the weak isospin asymmetry δμ =-20 MeV (The typical value of δμ in heavy ion collisions may be within this range, as estimated in [ 23 23 ]) 3 3 . The T – μ phase diagrams for varied <math id="M70"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> are shown in Fig. 1 Fig. 1 . For <math id="M71"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>=</mo><mn>2.0</mn><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> , Fig. 1(a) Fig. 1(a) shows two separate first-order phase boundaries, which correspond to the chiral transitions for the u and d quarks, respectively. For <math id="M72"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>=</mo><mn>0.6</mn><msub><mi>g</mi><mi>s</mi></msub></mrow></math> , Fig. 1(b) Fig. 1(b) shows that only one first-order chiral boundary emerges at the low temperature, but it splits into two lines at the relatively higher temperature. So there are still two critical endpoints. Further increasing <math id="M73"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> to 0.68 g s , Fig. 1(c) Fig. 1(c) displays that only one phase boundary appears. So we really observe that the two separate phase boundaries can be changed into one by the non-anomaly flavor-mixing induced by the mismatched vector interactions. The above calculation for δμ =-20 MeV is further extended to a fixed moderate coupling <math id="M76"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>=</mo><mn>0.4</mn><msub><mi>g</mi><mi>s</mi></msub></mrow></math> . The phase diagrams for varied <math id="M77"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> with <math id="M78"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>></mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> are shown in Fig. 2 Fig. 2 , which is still analogous to Fig. 1 Fig. 1 . In contrast to Fig. 1 Fig. 1 , a stronger <math id="M79"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> is required for the conversion of the two phase transitions into one due to the enlarged <math id="M80"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> . Fig. 2 Fig. 2 also shows that the chiral transition is first softened and then strengthened with <math id="M81"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> . By comparison, the chiral transition is always weakened with the increase of <math id="M82"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> . So for the weak isospin asymmetry, Figs. 1 Figs. 1 and 2 2 show that the chiral transition separation can be removed by the mismatched vector interactions, even without the instanton induced flavor-mixing. Actually, all the two sets of phase diagrams in Figs. 1 Figs. 1 and 2 2 are quite similar to Figs. 2 Figs. 2 in Ref. [ 23 23 ] obtained by changing the α . In this sense, the non-anomaly flavor-mixing due to the vector coupling difference plays the similar role as the KMT interaction. However, Figs. 1 Figs. 1 and 2 2 indicate that <math id="M85"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> must be much stronger than <math id="M86"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> for turning the two chiral transitions into one: <math id="M87"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> is at least twice as strong as <math id="M88"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> to remove the separation. Of course, the fate of the separate chiral transitions depends on not only the vector coupling difference, but also the magnitudes of <math id="M89"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> and <math id="M90"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> . Here we do not show the results for <math id="M91"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>></mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> with a fixed strong <math id="M92"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> since in this case only crossover transition appears. On the contrary, we do not find the coincidence of the detached phase boundaries for <math id="M93"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo><</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> . In Fig. 3 Fig. 3 , we show the phase diagrams for δμ =-20 MeV with varied <math id="M94"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> and fixed coupling <math id="M95"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>=</mo><mn>0.2</mn><msub><mi>g</mi><mi>s</mi></msub></mrow></math> . We see that the two separate phase boundaries get farther rather than closer with the increase of | δg v | for <math id="M96"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo><</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> , which is quite different from what shown in Figs. 1 Figs. 1 and 2 2 . The reason can be traced back to Eqs. (14) and (15). First, according to Eq. (15), the <math id="M99"><mrow><mo>|</mo><msub><msup><mi>μ</mi><mo>′</mo></msup><mi>I</mi></msub><mo>|</mo></mrow></math> is explicitly less than the <math id="M100"><mrow><mo>|</mo><msub><mi>μ</mi><mi>I</mi></msub><mo>|</mo></mrow></math> since the signs of μ I and <math id="M101"><mrow><mo>−</mo><mn>2</mn><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo stretchy="false">(</mo><msub><mi>ρ</mi><mi>u</mi></msub><mo>−</mo><msub><mi>ρ</mi><mi>d</mi></msub><mo stretchy="false">)</mo></mrow></math> in <math id="M102"><mrow><msub><msup><mi>μ</mi><mo>′</mo></msup><mi>I</mi></msub></mrow></math> are different for <math id="M103"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>></mo><mn>0</mn></mrow></math> . So for <math id="M104"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo>></mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> with a fixed <math id="M105"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> , increasing <math id="M106"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> implies not only the enhancement of the flavor-mixing but also the reduction of <math id="M107"><mrow><mo>|</mo><msub><mi>μ</mi><msup><mi>I</mi><mo>′</mo></msup></msub><mo>|</mo></mrow></math> . This is why the two phase boundaries approach each other with <math id="M108"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> , as shown in Figs. 1 Figs. 1 and 2 2 . Second, near the left side of the right phase boundary, the ρ d is remarkably larger than the ρ u because of the significant suppression of the d quark mass; but around the left side of the left phase boundary, the difference between the ρ d and ρ u is relatively small. So for <math id="M109"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>></mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> , the flavor-mixing term <math id="M110"><mrow><mo>−</mo><mo stretchy="false">(</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>−</mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo stretchy="false">)</mo><msub><mi>ρ</mi><mi>d</mi></msub></mrow></math> in <math id="M111"><mrow><msub><mi>μ</mi><msup><mi>u</mi><mo>′</mo></msup></msub></mrow></math> impacts the right phase boundary more significantly in contrast to what the corresponding term <math id="M112"><mrow><mo>−</mo><mo stretchy="false">(</mo><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>−</mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup><mo stretchy="false">)</mo><msub><mi>ρ</mi><mi>u</mi></msub></mrow></math> in <math id="M113"><mrow><msub><mi>μ</mi><msup><mi>d</mi><mo>′</mo></msup></msub></mrow></math> does on the left phase boundary, according to Eq. (14). This is why the right phase boundary moves more rapidly towards the higher μ with <math id="M114"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> in contrast to the left one, as shown in Fig. 3 Fig. 3 . If <math id="M115"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> or/and <math id="M116"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> are strong enough, the first-order chiral transition will change into crossover and it would be no critical point. Owning to vector interactions, it is possible that one of the two phase boundaries first disappears while the other one still remains with the change of vector interactions (In contrast, the two critical endpoints always appear at the same temperature in Refs. [ 11 11 , 10 10 ]). Such a case is really observed in Fig. 3 Fig. 3 for very strong vector interaction <math id="M117"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>=</mo><mn>1.0</mn><msub><mi>g</mi><mi>s</mi></msub></mrow></math> . Actually, our numerical study suggests that the emergence of only one critical endpoint via this manner does not require very strong vector interaction when the weak KMT interaction is included. Here only the weak isospin asymmetry is considered because | μ I | is small in heavy-ion collisions. On the other hand, the quark matter may appear in the core of neutron star. In such a case, the magnitude of the difference between the u and d quark chemical potentials may be as large as 100 MeV due to the constraint of charge neutrality 4 4 . For the strong isospin asymmetry with a relatively large | μ I |, we find that the separation of the chiral transition can not be removed by the non-anomaly flavor-mixing without considering the axial anomaly. However, the similar phase diagram as Fig. 3 Fig. 3 is still observed for a proper choice of <math id="M118"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> and <math id="M119"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> . V. DISCUSSION AND CONCLUSION 1 We have studied the influence of vector interactions with different coupling constants in the isoscalar and isovector channels on the possible separation of the chiral transition under the isospin asymmetry in a two-flavor NJL model, where the U (1) A symmetry is assumed to be restored effectively near the phase boundary. We first show that, besides the argument from the empirically different nucleon and vector-meson couplings [ 33 33 ], the one-gluon exchange type interaction can also give rise to unequal vector interactions with <math id="M120"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo>></mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> in the MFA when including the Fock contribution. On the other hand, by extending the work [ 24 24 ] to finite μ I , we obtain the quite different vector coupling difference with <math id="M121"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup><mo><</mo><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> from the constraints of lattice chiral curvatures at zero/small quark chemical potentials. We demonstrate that, similar to the mass-mixing induced by the KMT interaction, the density-mixing of two flavors is produced owning to the mismatched vector interactions. The role of the non-anomaly flavor-mixing on the chiral phase transition is investigated under the condition with weak isospin asymmetry. We find that to convert the two separate chiral transitions into one, <math id="M122"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>v</mi></msubsup></mrow></math> must be significantly stronger than <math id="M123"><mrow><msubsup><mi>g</mi><mi>v</mi><mi>s</mi></msubsup></mrow></math> without the axial anomaly. In this situation, the non-anomaly flavor-mixing induced by vector interactions impacts the separation of chiral phase transitions in the similar way as the anomaly one induced by instantons: the two detached phase boundaries get closer first and then coincide with the enhancement of the flavor-mixing. Note that recently the Polyakov-Loop extended NJL model has been extensively used to investigate the thermal and dense properties of QCD. We stress that introducing the Polyakov-Loop dynamics does not qualitatively change our main conclusions. 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