Influences of magnetic field on the coexistence of diquark and chiral condensates in the Nambu【-逻*辑*
本站小编 Free考研考试/2022-01-02
Xiao-Bing Zhang(张小兵)1, Fu-Ping Peng(彭富平),1, Yun-Ben Wu(吴云奔)1, Yi Zhang(张一)21 School of Physics, Nankai University, Tianjin 300071, China 2 Department of Physics, Shanghai Normal University, Shanghai 200230, China
Abstract In this paper, we study the influences of magnetic fields on the coexistence of diquark and chiral condensates in an extended Nambu–Jona–Lasinio model with QCD axial anomaly, as it relates to color-flavor-locked quark matter. Due to the coupling of rotated-charged quarks to magnetic fields, diquark condensates become split, and the coexistence region is thus superseded in favor of a specific diquark Bose–Einstein condensation (BEC), denoted as the BECI phase. For strong magnetic fields, we find that the BECI transition is pushed to larger quark chemical potentials. The effect of magnetic catalysis tends to disrupt the BEC–BCS (Bardeen–Cooper–Schrieffer) crossover predicted in previous works. For intermediate fields, the effect of inverse magnetic catalysis is observed, and the axial-anomaly-induced phase structure is essentially unchanged. Keywords：Nambu–Jona–Lasinio model;diquark Bose–Einstein condensation;magnetic catalysis;inverse magnetic catalysis;axial-anomaly;coexistence of diquark and chiral condensates;magnetic field
PDF (579KB)MetadataMetricsRelated articlesExportEndNote|Ris|BibtexFavorite Cite this article Xiao-Bing Zhang(张小兵), Fu-Ping Peng(彭富平), Yun-Ben Wu(吴云奔), Yi Zhang(张一). Influences of magnetic field on the coexistence of diquark and chiral condensates in the Nambu–Jona–Lasinio model with axial anomaly. Communications in Theoretical Physics[J], 2020, 72(11): 115302- doi:10.1088/1572-9494/aba25b
1. Introduction
The phases of quantum chromodynamics (QCD) and the phase structure of strongly interacting matter have attracted great interest for many years. In addition to hadronic matter and quark gluon plasma, Bardeen–Cooper–Schrieffer (BCS) diquark pairing leads to quark color superconductivity at low temperatures, and high baryon density, see e.g. [1 –3] for reviews. The three-color and three-flavor case corresponds to color-flavor-locked (CFL) matter, and is believed to be the ground state of QCD for sufficiently high densities [4]. The condensation of quark–antiquark pairs $\langle \bar{q}q\rangle $, specifically chiral condensate, breaks chiral symmetry in vacuum and in hadronic matter. In CFL matter, the condensation of diquark pairs $\langle {qq}\rangle $ continues to break chiral symmetry, even though $\langle \bar{q}q\rangle $ disappears at high densities. Therefore, a coexistence of $\langle \bar{q}q\rangle $ and $\langle {qq}\rangle $ should be expected at moderate density and low temperature. Correspondingly, rich phase structures may also exist in the coexistence regime.
This theoretical possibility has been previously proposed in [5, 6]. By considering the QCD axial anomaly within a Ginzburg–Landau framework, the interplay between chiral and diquark condensates leads to a new critical point in the phase diagram of three-flavor dense matter. This type of coexistence and the critical phenomenon has also been investigated in relation to phenomenological quark models. In [7], chiral-diquark interplay was introduced via six-quark effective interactions, and a Nambu–Jona–Lasinio (NJL) model with axial anomaly was employed. As pointed out in [7], a coexistence phase regime exists between a chiral-broken phase and a BCS phase of CFL matter. Moreover, the critical phenomenon takes place in the sense that the Bose–Einstein condensation (BEC) of diquark ‘molecules’ emerges firstly, followed by a BEC–BCS crossover. In more realistic situations, flavor asymmetry, the confinement effect, and color neutrality may be taken into account in the NJL model [8 –10].
Notably, quark matter is subject to magnetic fields, and the influence of magnetic fields is usually nontrivial. Strong magnetic fields of the order of ${eB}\sim {m}_{\pi }^{2}\approx {10}^{18}{\rm{G}}$ can be produced in off-central heavy ion collisions for the collision energy $\sqrt{s}=200\,\mathrm{GeV}$ in the Relativistic Heavy Ion Collider (RHIC), with even higher field strengths of ${eB}\sim {10}^{20}\,{\rm{G}}$ available via the Large Hadron Collider (LHC). These fields are also believed to appear in the interior regions of compact stars. It has been widely realized that the magnetic field significantly affects the QCD phase diagram, (see e.g. [11, 12]). At vanishing quark chemical potential μ, lattice calculations have been employed to study the influence of magnetic fields on chiral condensate. Magnetic catalysis was initially confirmed [13 –15], with inverse magnetic catalysis being found by virtue of more recent calculations [11, 12, 16]. At large μ and small temperature T, on the other hand, color superconductivity with diquark condensate is dominant. In this case, a linear combination of electromagnetic and color gauge symmetries, denoted as $U{(1)}_{\widetilde{Q}}$, remains unbroken [4, 17]. This is known as the rotated electromagnetic mechanism. The presence of a rotated magnetic field leads to the splitting of diquark condensates in the BCS phase of CFL matter, which has been studied in relation to NJL-type models without axial anomaly [18 –21]. However, the situation remains unclear as to whether the coexistence induced by axial anomaly actually exists. An investigation into the influence of magnetic fields on phase structure in coexistence regimes is a highly significant topic.
In this paper, we will introduce an applied magnetic field B, being the rotated field corresponding to $U{(1)}_{\widetilde{Q}}$, in the NJL description of three-flavor dense matter with axial-anomaly. We are particularly interested in the following areas: (i) how coexistence is modified by magnetic fields, and (ii) the consequence of the magnetic fields in terms of BEC. Here, area (i) may be regarded as an extension of [7]. We will compare phase diagrams with and without the presence of a magnetic background field. Due to the splitting of diquark condensates, the different kinds of BEC, and the possible consequences, are studied in detail. Area (ii) concentrates on a specific BEC, associated with (rotated) charged pairing, denoted as BECI . The emergence of BECI might be considered as this paper’s main difference from earlier work. As we will see below, this factor significantly changes the phase structure for strong magnetic fields, leading to an effect of inverse magnetic catalysis for intermediate fields.
The paper is organized as follows: the model under investigation is introduced in section 2 . Having defined the new phases, we present the numerical results of gap changes and phase diagrams in section 3 . In section 4, we explain the main results and the emergence of BECI in detail. Finally, our conclusions are drawn in section 5 .
2. Model and formalism
As mentioned in section 1, in general, there are two types of order parameters. At low baryon densities, chiral symmetry becomes broken, and the relevant order parameter is the chiral condensate $\chi =\langle \bar{q}q\rangle $ . At high densities and zero temperature, color superconducting matter becomes possible where diquark condensates occur in the qq channel. For CFL matter, these are generally defined as$ \begin{eqnarray}{s}_{{{AA}}^{{\prime} }}= \langle {q}^{{\rm{T}}}C{\gamma }_{5}{\tau }_{A}{\lambda }_{{A}^{{\prime} }}q \rangle ,\end{eqnarray}$ in any representation of the color-antitriplet and the flavor-antitriplet [4, 22]. Here, τA and ${\lambda }_{A^{\prime} }$, together with $A^{\prime} \,=A=2,5,7$, denote the three antisymmetric generators of SU (3)F and SU (3)C, respectively. For instance, the order parameter s55 describes the pairing of quarks with flavors d and s, and colors g and b . A list of these order parameters can be found in [22, 23].
In order to study the coexistence of chiral and diquark condensates, our starting point is an extended NJL Lagrangian:$ \begin{eqnarray}{ \mathcal L }=\bar{q}({\rm{i}}{\gamma }^{\mu }{{\rm{\partial }}}_{\mu }+{\gamma }_{0}\mu -{m}_{q})q+{{ \mathcal L }}^{(4)}+{{ \mathcal L }}^{(6)},\end{eqnarray}$ where ${{ \mathcal L }}^{(4)}$ denotes four-quark interaction, and ${{ \mathcal L }}^{(6)}$ is the six-quark interaction originating from the QCD axial anomaly. Using the order parameters defined above, these interaction terms can be derived at the mean-field level (see [7] for details). Since both the $\bar{q}q$ interaction and the qq interaction are considered, the mean-field form of ${{ \mathcal L }}^{(4)}$ consists of two parts, i.e.$ \begin{eqnarray}{{ \mathcal L }}_{\chi }^{(4)}=4G\chi \bar{q}q-6G{\chi }^{2},\end{eqnarray}$$ \begin{eqnarray}{{ \mathcal L }}_{s}^{(4)}=\displaystyle \sum _{A,A^{\prime} =2,5,7}\displaystyle \frac{1}{3}H[{s}_{{AA}^{\prime} }^{* }({q}^{{\rm{T}}}C{\gamma }_{5}{\tau }_{A}{\lambda }_{A^{\prime} }q)+{\rm{h}}.{\rm{c}}.]-H| {s}_{{AA}^{\prime} }{| }^{2}.\end{eqnarray}$ Obviously, G and H denote the coupling parameters in the $\bar{q}q$ and qq channels, respectively.
In addition, the form of the six-quark interaction consists of two parts. The standard Kobayashi–Maskawa–’t Hooft interaction gives the mean-field term$ \begin{eqnarray}{{ \mathcal L }}_{\chi }^{\left(6\right)}=-2K{\chi }^{2}\bar{q}q+4K{\chi }^{3},\end{eqnarray}$ where K is the coupling parameter, having the dimension (mass)−5 . The effective interaction between $\bar{q}q$ and qq pairings has previously been introduced in [5, 6]. By following the process outlined in [7], the mean-field form reads$ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{\chi s}^{\left(6\right)} & = & -\displaystyle \sum _{A,A^{\prime} =2,5,7}\displaystyle \frac{K^{\prime} }{12}| {s}_{{AA}^{\prime} }{| }^{2}\bar{q}q\\ & & -\displaystyle \frac{K^{\prime} }{12}\chi [{s}_{{AA}^{\prime} }^{* }({q}^{{\rm{T}}}C{\gamma }_{5}{\tau }_{A}{\lambda }_{A^{\prime} }q)+{\rm{h}}.{\rm{c}}.]+\displaystyle \frac{K^{\prime} }{2}| {s}_{{AA}^{\prime} }{| }^{2}\chi .\end{array}\end{eqnarray}$ Here, $K^{\prime} $ behaves as the coupling parameter, describing the interplay between chiral and diquark condensates. As long as the selected value is appropriately large, coexistence becomes possible (as discussed in [7]).
2.1. Diquark condensate splitting and interacting potential
It is well known that so-called rotated electromagnetic mechanism plays a key role in CFL superconducting matter [4]. The essentially unbroken symmetry is defined as the subgroup $U{(1)}_{\widetilde{Q}}$, with a rotated electric charge $\widetilde{Q}=Q\,\times {{\bf{1}}}_{3}+{{\bf{1}}}_{3}\times {\lambda }_{8}/\sqrt{3}$, where Q and λ8 denote the electric charge matrix, and the eighth Gell–Mann matrix, respectively. In this sense, the rotated electromagnetic field becomes a linear combination of the electromagnetic field and the eighth gluon field. Furthermore, the paired quarks are either rotated-charge neutral, or charged with $\widetilde{Q}=\pm 1$ (see table 1 ). In order to describe quark matter subject to magnetic fields, the derivative in the Lagrangian(2 ) shall be replaced by ${{\rm{\partial }}}_{\mu }-{\rm{i}}\tilde{e}{\tilde{A}}_{\mu }$ . Due to the large coupling of QCD, the rotated field $\widetilde{A}$ largely consists of a typical electromagnetic field. At the same time, the unit of rotated charge $\widetilde{e}$ is expressed approximately as e . In the Landau gauge, therefore, the rotated magnetic field can be described as an external background field, B .
Table 1. Table 1.Classification of pairing type, gaps, and dispersion relations.
With the introduction of this type of applied field, the diquark condensates behave differently from each other. As observed in [18, 20, 24, 25], the order parameters become split$ \begin{eqnarray}s={s}_{55},\quad {s}_{B}={s}_{22}={s}_{77},\end{eqnarray}$ rather than exhibiting the degenerated order parameter s22 =s55 =s77 found in the absence of a magnetic field. Having expanded the order parameters to χ, s and sB, we revisit the forms of ${{ \mathcal L }}^{(4)}$ and ${{ \mathcal L }}^{(6)}$ . In the $\bar{q}q$ channel, it is obvious that equations (3 ) and (5 ) hold unchanged. Due to the ansatz equation (7 ), the other interaction terms then read as follows:$ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{s}^{(4)} & = & \displaystyle \frac{1}{3}H[{s}_{B}^{* }({q}^{{\rm{T}}}C{\gamma }_{5}{\tau }_{2}{\lambda }_{2}q)+{\rm{h}}.{\rm{c}}.]-H| {s}_{B}{| }^{2}\\ & & +\displaystyle \frac{1}{3}H[{s}^{* }({q}^{{\rm{T}}}C{\gamma }_{5}{\tau }_{5}{\lambda }_{5}q)+{\rm{h}}.{\rm{c}}.]-H| s{| }^{2}\\ & & +\displaystyle \frac{1}{3}H[{s}_{B}^{* }({q}^{{\rm{T}}}C{\gamma }_{5}{\tau }_{7}{\lambda }_{7}q)+{\rm{h}}.{\rm{c}}.]-H| {s}_{B}{| }^{2},\end{array}\end{eqnarray}$ and$ \begin{eqnarray}\begin{array}{rcl}{{ \mathcal L }}_{\chi s}^{\left(6\right)} & = & -\displaystyle \frac{K^{\prime} }{12}| {s}_{B}{| }^{2}\bar{q}q-\displaystyle \frac{K^{\prime} }{12}\chi [{s}_{B}^{* }({q}^{{\rm{T}}}C{\gamma }_{5}{\tau }_{2}{\lambda }_{2}q)+{\rm{h}}.{\rm{c}}.]\\ & & -\displaystyle \frac{K^{\prime} }{12}| s{| }^{2}\bar{q}q-\displaystyle \frac{K^{\prime} }{12}\chi [{s}^{* }({q}^{{\rm{T}}}C{\gamma }_{5}{\tau }_{5}{\lambda }_{5}q)+{\rm{h}}.{\rm{c}}.]\\ & & -\displaystyle \frac{K^{\prime} }{12}| {s}_{B}{| }^{2}\bar{q}q-\displaystyle \frac{K^{\prime} }{12}\chi [{s}_{B}^{* }({q}^{{\rm{T}}}C{\gamma }_{5}{\tau }_{7}{\lambda }_{7}q)+{\rm{h}}.{\rm{c}}.]\\ & & +\displaystyle \frac{K^{\prime} }{2}| s{| }^{2}\chi +K^{\prime} | {s}_{B}{| }^{2}\chi ,\end{array}\end{eqnarray}$ at the mean-field level. Collecting these terms, we give the interacting potential as a function of χ, s and sB, i.e.$ \begin{eqnarray}U=6G{\chi }^{2}-4K{\chi }^{3}+\left(H-\displaystyle \frac{K^{\prime} }{2}\chi \right)(| s{| }^{2}+2| {s}_{B}{| }^{2}).\end{eqnarray}$
On the other hand, three order parameters correspond to three ‘gaps’. In the qq channel, the color superconducting gaps Δ and ${{\rm{\Delta }}}_{B}$ are given as$ \begin{eqnarray}{\rm{\Delta }}=-2H^{\prime} s,\end{eqnarray}$$ \begin{eqnarray}{{\rm{\Delta }}}_{B}=-2H^{\prime} {s}_{B}.\end{eqnarray}$ Here, an effective diquark coupling$ \begin{eqnarray}H^{\prime} =H+\displaystyle \frac{1}{4}K^{\prime} | \chi | ,\end{eqnarray}$ has been defined for later convenience. Moreover, the constituent quark mass might be regarded as a generalized gap in the $\bar{q}q$ channel. In the chiral limit, this is given as$ \begin{eqnarray}M=-4\left(G-\displaystyle \frac{1}{8}K\chi \right)\chi +\displaystyle \frac{1}{12}K^{\prime} | s{| }^{2}+\displaystyle \frac{1}{6}K^{\prime} | {s}_{B}{| }^{2}.\end{eqnarray}$
2.2. Thermodynamic potential
As is generally the case, the thermodynamic potential of the system shall be composed of the interacting potential U, in addition to the contribution from quarks. In order to obtain the latter form, a standard treatment is to consider a trace such as $-T\sum \int {{\rm{d}}}^{3}p\mathrm{Tr}\ \mathrm{ln}[{S}^{-1}/T]$ with the inverse quark propagator$ \begin{eqnarray}\begin{array}{l}{S}^{-1}(p)\\ =\,\left(\begin{array}{cc}{\gamma }_{\mu }{p}^{\mu }+\mu {\gamma }_{0}-M & \displaystyle \sum _{A,A^{\prime} =2,5,7}H^{\prime} {s}_{{AA}^{\prime} }{\gamma }_{5}{\tau }_{A}{\lambda }_{A^{\prime} }\\ -\displaystyle \sum _{A,A^{\prime} =2,5,7}H^{\prime} {s}_{{AA}^{\prime} }^{* }{\gamma }_{5}{\tau }_{A}{\lambda }_{A^{\prime} } & {\gamma }_{\mu }{p}^{\mu }-\mu {\gamma }_{0}-M\end{array}\right),\end{array}\end{eqnarray}$ as defined in the bispinor Nambu–Gorkov space. Now that diquark condensates occur in the color and flavor space, the off-diagonal elements of equation (15 ) require to be expressed as 9-dimensional (color×flavor) block matrices (see equations (A24) and (A26) in [25] for details). In the presence of a magnetic field, it transpires that the pairing pattern may be classified into three types: neutral, charged, and mixed parings. For each type, the rotated charges of paired quarks, the color superconducting gaps, and the quasiparticle dispersion relations are as summarized in table 1 . Whereas the gaps correspond to the values of Δ and ΔB defined above, and their combinations, the dispersion relation for quasiquarks is derived from the vanishing determinant det(S−1 ) in the 2×9-dimension space. As shown in table 1, there are three types of dispersion relations, εn, εc and εm, whose forms will be given below.
Based on the three pairing types, we evaluate the corresponding thermodynamic potential contributions from quarks at the zero temperature limit, respectively, below.
2.2.1. Charged pairing type
As shown in table 1, this pairing type describes rs –bu and/or gu –rd pairing. Using the gap ΔB, the dispersion relation reads ${\epsilon }^{c}=\sqrt{{\left({E}_{B}\pm \mu \right)}^{2}+{{\rm{\Delta }}}_{B}^{2}}$, with the single-particle energy$ \begin{eqnarray}{E}_{B}=\sqrt{{p}_{3}^{2}+{M}^{2}+2{neB}},\end{eqnarray}$ where the third momentum p3 and the Landau level index n =0, 1, 2, ⋯are introduced. Moreover, the integral over the three-momentum is substituted by the summation over the Landau level:$ \begin{eqnarray}2\int \displaystyle \frac{{{\rm{d}}}^{3}p}{{\left(2\pi \right)}^{3}}\to \displaystyle \frac{{eB}}{8{\pi }^{2}}\displaystyle \sum _{n}^{\infty }(2-{\delta }_{n0}){\int }_{-\infty }^{+\infty }{\rm{d}}{p}_{3}.\end{eqnarray}$ In the numerical calculation, we will use the maximum ${n}_{\max }=\mathrm{Int}[{{\rm{\Lambda }}}^{2}/2{eB}]$ as the number of completely occupied Landau levels. Correspondingly, the thermodynamic potential for the charged pairing type can be expressed as$ \begin{eqnarray}{{\rm{\Omega }}}_{q}^{c}=-8\displaystyle \sum _{\pm }\displaystyle \frac{{eB}}{8{\pi }^{2}}\displaystyle \sum _{n=0}^{{n}_{\max }}(2-{\delta }_{n0})\int {h}_{{\rm{\Lambda }},B}^{n}{\rm{d}}{p}_{3}| {\epsilon }^{c}| ,\end{eqnarray}$ rather than the form $\sim {\sum }_{\pm }\int {{\rm{d}}}^{3}p| \epsilon | $ used in the absence of a magnetic field.
In equation (18 ), the form factor ${h}_{{\rm{\Lambda }},B}^{n}$ is used. It is known that, for nonzero magnetic fields, hard regularization with a simple step function leads to strong nonphysical oscillations [26]. In the literature, various smooth form factors (e.g. the Gaussian, Lorentzian, and Wood–Saxon form factors) have previously been considered in order to improve the situation. Here, we employ the Lorentzian form factor$ \begin{eqnarray}{h}_{{\rm{\Lambda }},B}^{n}={\left[1+{\left(\displaystyle \frac{\sqrt{{p}_{3}^{2}+2{neB}}}{{\rm{\Lambda }}}\right)}^{N}\right]}^{-1},\end{eqnarray}$ where the momentum cutoff Λ denotes a model parameter, and N denotes the Lorentzian index. The numerical calculations for magnetized color superconducting matter show that this regularization scheme enables the removal of nonphysical oscillations as much as possible (see, e.g. [27, 28]).
2.2.2. Neutral and mixed types
These two cases are straightforward, since the paired quarks are neutral and do not couple with the magnetic field directly. For the neutral type (i.e. the bd –gs pairing), only the order parameter s is involved. The dispersion relation is ${\epsilon }^{n}=\sqrt{{\left(E\pm \mu \right)}^{2}+{{\rm{\Delta }}}^{2}}$, with the single-particle energy $E=\sqrt{{p}^{2}+{M}^{2}}$ . The thermodynamic potential for this type is obtained as$ \begin{eqnarray}{{\rm{\Omega }}}_{q}^{n}=-6\displaystyle \sum _{\pm }\int {h}_{{\rm{\Lambda }}}\displaystyle \frac{{{\rm{d}}}^{3}p}{{\left(2\pi \right)}^{3}}| {\epsilon }^{n}| ,\end{eqnarray}$ where the Lorentzian form factor is reduced to ${h}_{{\rm{\Lambda }}}\,={\left[1+{\left(\tfrac{p}{{\rm{\Lambda }}}\right)}^{N}\right]}^{-1}$ .
For the mixed type (i.e. ru –gd –bs pairing), two order parameters, s and sB, take effect simultaneously. The color superconducting gaps are denoted by combinations of Δ and ΔB, (see e.g. [19, 25]):$ \begin{eqnarray}{{\rm{\Delta }}}_{1}=\displaystyle \frac{1}{2}(\sqrt{{{\rm{\Delta }}}^{2}+8{{\rm{\Delta }}}_{B}^{2}}+{\rm{\Delta }}),\quad {{\rm{\Delta }}}_{2}=\displaystyle \frac{1}{2}(\sqrt{{{\rm{\Delta }}}^{2}+8{{\rm{\Delta }}}_{B}^{2}}-{\rm{\Delta }}).\end{eqnarray}$ Correspondingly, the dispersion relations have the following forms:$ \begin{eqnarray}{\epsilon }_{1}^{m}=\sqrt{{\left(E\pm \mu \right)}^{2}+{{\rm{\Delta }}}_{1}^{2}},\quad {\epsilon }_{2}^{m}=\sqrt{{\left(E\pm \mu \right)}^{2}+{{\rm{\Delta }}}_{2}^{2}},\end{eqnarray}$ and the thermodynamic potential becomes$ \begin{eqnarray}{{\rm{\Omega }}}_{q}^{m}=-2\displaystyle \sum _{\pm }\int {h}_{{\rm{\Lambda }}}\displaystyle \frac{{{\rm{d}}}^{3}p}{{\left(2\pi \right)}^{3}}{\epsilon }_{1}^{m}-2\displaystyle \sum _{\pm }\int {h}_{{\rm{\Lambda }}}\displaystyle \frac{{{\rm{d}}}^{3}p}{{\left(2\pi \right)}^{3}}{\epsilon }_{2}^{m}.\end{eqnarray}$
To summarize, the total thermodynamic potential reads$ \begin{eqnarray}{\rm{\Omega }}={{\rm{\Omega }}}_{q}^{c}+{{\rm{\Omega }}}_{q}^{n}+{{\rm{\Omega }}}_{q}^{m}+U.\end{eqnarray}$ Here, we have dropped out the electromagnetic field contribution $\tfrac{{B}^{2}}{2}$ in the Landau gauge.
As the global minimum of the thermodynamic potential(24 ), the order parameters s, sB, and χ are determined by$ \begin{eqnarray}\displaystyle \frac{\partial {\rm{\Omega }}}{\partial s}=0,\quad \displaystyle \frac{\partial {\rm{\Omega }}}{\partial {s}_{B}}=0,\quad \displaystyle \frac{\partial {\rm{\Omega }}}{\partial \chi }=0,\end{eqnarray}$ self-consistently. These equations are highly coupled, and must be solved numerically. Clearly, the gaps Δ, ΔB, and the mass M must be obtained via equations (11 ), (12 ), and (14 ).
3. Numerical results
In the extended model with axial anomaly, G, H, K, and $K^{\prime} $ are treated as independent model parameters. The chiral couplings G and K, as well as the momentum cutoff Λ are chosen by fitting mesonic properties in the QCD vacuum, while the diquark coupling parameter H is chosen in relation to the color superconducting gaps (see, e.g. [22, 23]). As for the chiral-diquark coupling parameter $K^{\prime} $, its value (to be precise, the ratio $K^{\prime} /K$ ) is required for the emergence of coexistence. For our purposes, we follow the parameter set I, as used in [7], and adopt the qualified value $K^{\prime} =4.2K$ (unless otherwise stated). In addition, note that the Lorentzian form factor where N =5, i.e. the so-called Lor5 regularization, is employed.
In the absence of a magnetic field, the coexistence of chiral and diquark condensates manifests itself as the BEC phase of diquark ‘molecules’. This type of phase is an intermediate state between the Nambu–Goldstone (NG) phase, with chiral symmetry breaking, and the BCS phase of color superconducting matter. In the presence of magnetic fields, we must introduce various types of Bose–Einstein condensed phases. To be specific, we refer to the phase where ${s}_{B}\ne 0$ but s =0 and $\chi \ne 0$ as BECI, while that where $s\ne 0$, sB =0 and $\chi \ne 0$ is denoted as BECII . The normal BEC phase occurs where $s\ne 0$, ${s}_{B}\ne 0$ and $\chi \ne 0$ .
We first calculate three ‘gaps’ for intermediate magnetic fields. Figure 1 shows a typical result, calculated at ${eB}=4.6{m}_{\pi }^{2}$ . At the quark chemical potential ${\mu }_{c}^{{\rm{I}}}\approx 313$ MeV, BECI emerges, with the appearance of sB and thus ΔB . Thereafter ${\mu }_{c}^{{\rm{I}}}$ denotes the critical value of BECI transition. It is found (see figure 1 ) that Δ is not permitted to appear until the critical value of BEC transition, μc ≈327 MeV, is reached. Thus, coexistence actually emerges in such way that BECI appears first, and is followed by a normal BEC phase. The BEC phase is terminated as the chemical potential increases up to μX ≈350 MeV, which satisfies the condition M =μ (see the dotted vertical line in figure 1 ). In the vicinity of μX, the order parameters are observed to be continuous, and there is a smooth crossover from BEC to BCS.
Figure 1.
New window|Download| PPT slide Figure 1.Constituent quark mass M (dotted line), color superconducting gaps ΔB (red solid line) and Δ (blue dash–dotted line) as functions of μ for ${eB}=4.6{m}_{\pi }^{2}$ . The momentum cutoff is ${\rm{\Lambda }}=602.3\,\mathrm{MeV}$, and the coupling constants are selected as $G=1.835/{{\rm{\Lambda }}}^{2}$, H =1.74/Λ2, K =12.36/Λ5, and $K^{\prime} =4.2K$ .
A better illustration of the phase structure can be found in the μ -$K^{\prime} $ plane. With varying strengths of $K^{\prime} $, a schematic phase diagram is plotted in figure 2 . For small $K^{\prime} $, the NG and BCS phase regimes are simply separated by a first-order chiral phase transition (thick solid line). When the appropriate values of $K^{\prime} $ are considered, the interplay between chiral and diquark condensates takes effect. As shown in figure 2, the BECI regime appears across a second-order transition (thin red line). The ${\mathrm{BEC}}_{{\rm{I}}}$ transition joins the phase boundary (thick solid line) at point Q1 . The intersection point is regarded as a critical end point, since the conventional chiral transition is no longer valid there. Similarly, the BEC regime follows across the dash–dotted blue line. The intersection of BEC transition and the thick solid line produces another critical end point, Q2 . Furthermore, the phase boundary given by the thick solid line is terminated at point P . There, the order parameters remain continuous, and thus a smooth BEC–BCS crossover takes place at the critical point P . Physically, the three critical points emanate from the chiral-diquark interplay induced by the anomaly $K^{\prime} $ . The phase diagram in figure 2 is essentially similar to the previous zero-field result [7]. The key difference lies in the fact that here, point Q splits into Q1 and Q2, corresponding to the splitting of diquark condensates induced by the magnetic field. Moreover, we note that the present mass is larger compared with that given in [7]. This point is mainly associated with the magnetic effect on the chiral condensate.
Figure 2.
New window|Download| PPT slide Figure 2.Phase diagram in the μ -$K^{\prime} $ plane for an intermediate magnetic field. The NG–BCS boundary is shown by a thick (black) solid line, which is a first-order transition for small $K^{\prime} $ . The emergence of BECI and BEC are shown by the thin (red) solid and dash–dotted (blue) lines, respectively. The BEC–BCS boundary is given by M =μ and is shown as a dotted (black) line. The three critical points, Q1, Q2, and P, are explained in the text.
With an increase in the magnetic field, significant changes in phase structure can appear. As the magnetic field strength approaches a threshold value, numerical calculation implies that only BECI exists between NG and BCS. Figure 3 shows the results for a relatively strong field, ${eB}=9.2{m}_{\pi }^{2}$ . Here, it can be seen that BECI emerges at ${\mu }_{c}^{{\rm{I}}}$ and is terminated at μX . In contrast to the weak-field case shown in figure 1, the $\mathrm{BEC}$ phase is now ‘squeezed out’ completely. The corresponding phase diagram is plotted in figure 4 . Here, point Q1 is still a critical end point, but point Q2, as given in figure 2, disappears from the phase diagram.
Figure 3.
New window|Download| PPT slide Figure 3.Constituent quark mass and color superconducting gaps for ${eB}=9.2{m}_{\pi }^{2}$ . The choice of parameters is consistent with figure 1 .
Figure 4.
New window|Download| PPT slide Figure 4.Phase diagram in the μ -$K^{\prime} $ plane for a strong magnetic field. Similar lines to figure 2 are employed. While Q1 remains a critical point, point P (the hollow circle) is no longer a critical point.
This, however, is not yet the whole story. While continuity remains at ${\mu }_{c}^{{\rm{I}}}$, the quark mass is no longer continuous at around μX . As shown in figure 3, discontinuous changes in mass and superconducting gaps occur at this chemical potential. Correspondingly, point P, marked by a hollow circle in figure 4, is merely an intersection of two lines. Due to the discontinuity around this point, a smooth BECI -BCS crossover does not take place. Thus it is unreasonable to regard point P as a critical point. This is the most significant difference from the previous results (including the weak-field result). The phenomenon occurring at point P can be better understood from the viewpoint of the symmetry breaking patterns of order parameters. In the situation of vanishing and/or weak magnetic fields, the pattern of diquark condensates is ${SU}{(3)}_{C}\times {SU}{(3)}_{L}\times {SU}{(3)}_{R}$$\times U{(1)}_{B}\times U{(1)}_{A}\,\to {SU}{(3)}_{C+L+R}$ . The color-flavor-locked pattern is similar, with the symmetry breaking of chiral condensate being ${SU}{(3)}_{L}\,\times {SU}{(3)}_{R}\to {SU}{(3)}_{L+R}$ . As stressed in [5, 6], the axial anomaly provides the chiral-diquark interplay, and leading eventually to the critical phenomenon shown in the phase diagram. In the BCS phase of magnetized CFL matter, however, it transpires that the symmetry may be further broken to ${SU}{(2)}_{C+L+R}$ (see, e.g. [19]). This phenomenon usually takes place when the magnitude of the magnetic field becomes relatively large. In this case, the symmetry breaking patterns of diquark and chiral condensates are no longer consistent with each other. Purely from the consideration of symmetry, the anomaly-induced critical phenomenon is cannot possibly remain unchanged at the boundary of the BCS phase. This explains why the expected critical phenomenon around point P becomes disrupted in the presence of a relatively strong magnetic background.
4. Discussions
Here, we further interpret the numerical results of the phase structure, and answer the questions raised in section 1 . For this purpose, we first address the criteria of various Bose–Einstein condensates.
As a second-order transition, the occurrence of BEC is generally determined by the Thouless criterion [29, 30]. Equally, the criterion may be derived from the thermodynamic potential of the system. In terms of the BECII transition, the relevant order parameter is s, and the paired quarks are rotated-charge neutral. We consider the thermodynamic potential in equation (24 ) as a Ginzburg–Landau (GL) expansion of the order parameter s . The criterion is then obtained from the vanishing coefficient in the quadratic GL term. Using this condition, the occurrence of BECII is determined by$ \begin{eqnarray}\displaystyle \frac{1}{H^{\prime} }=\displaystyle \frac{8}{{\pi }^{2}}\int {h}_{{\rm{\Lambda }}}\displaystyle \frac{{{Ep}}^{2}}{{E}^{2}-{\mu }^{2}}{\rm{d}}p,\end{eqnarray}$ which has a similar form to that used in [7].
The case of BECI is a little complicated. For the order parameter sB, both charged and neutral quarks are involved simultaneously. As illustrated in table. 1, these are associated with charged and mixed pairings, respectively. Taking the two types of contribution into account, the BECI criterion is also obtained from the vanishing GL coefficient. Its explicit form reads$ \begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{1}{H^{\prime} } & = & \displaystyle \frac{{eB}}{2{\pi }^{2}}\displaystyle \int \displaystyle \sum _{n=0}^{{n}_{\max }}\left(1-\displaystyle \frac{{\delta }_{n0}}{2}\right){h}_{{\rm{\Lambda }},B}^{n}\displaystyle \frac{{E}_{B}}{{E}_{B}^{2}-{\mu }^{2}}{\rm{d}}{p}_{3}\\ & & +\displaystyle \frac{4}{{\pi }^{2}}\displaystyle \int {h}_{{\rm{\Lambda }}}\displaystyle \frac{{{Ep}}^{2}}{{E}^{2}-{\mu }^{2}}{\rm{d}}p.\end{array}\end{eqnarray}$ The first term on the right-hand side comes from the charged quark contribution, so that a direct coupling to the magnetic field appears. The eB dependence and the Landau levels in this term will be further discussed below.
Using these criteria, we obtain the critical chemical potentials ${\mu }_{c}^{{\rm{I}}}$ and ${\mu }_{c}^{\mathrm{II}}$, and consequently calculate the critical value μc for BEC transition. Their magnetic field dependence is plotted in figure 5 . In addition, the chemical potential μX, describing the boundary of the Bose–Einstein condensed phase, is marked. These competing chemical potentials may be responsible for the phase structure obtained in section 3 . We first interpret the precedence of the BECI transition. With an increase in the magnetic field, it can be seen that ${\mu }_{c}^{{\rm{I}}}$ decreases, whereas ${\mu }_{c}^{\mathrm{II}}$ and μc increase in the interval $6{m}_{\pi }^{2}\lt {eB}\lt 10{m}_{\pi }^{2}$ . Since the value of ${\mu }_{c}^{{\rm{I}}}$ is always less than the others, the BECI transition precedes either the BECII or BEC transition. With regard to the threshold value mentioned in section 3, this corresponds to the magnetic field at which μc meets μX . It is shown in figure 5 that the threshold field is ${eB}\approx 9{m}_{\pi }^{2}$ . As long as the magnetic field is larger than it, BEC disappears from the phase diagram.
Figure 5.
New window|Download| PPT slide Figure 5.Magnetic field dependences of ${\mu }_{c}^{{\rm{I}}}$ (red solid line), ${\mu }_{c}^{\mathrm{II}}$ (green dashed line), μc (blue dash–dotted line), and μX (black dotted line), where $K^{\prime} =4.2K$ .
We now answer question (ii), originally raised in section 1 . Here, we pay particular attention to the effect of a magnetic field on the BECI transition, since this precedes the other transitions. In the limit of a strong magnetic field, it is known that the lowest Landau level contribution is dominant, and that dimensional reduction might take effect [13 –15]. In relation to the BECI transition, its criterion is given in equation (27 ). In the strong-field limit, the first term in the right-hand side of equation (27 ) is reducible as follows:$ \begin{eqnarray}\displaystyle \frac{{eB}}{4{\pi }^{2}}\int {h}_{{\rm{\Lambda }},B}^{n=0}\displaystyle \frac{\sqrt{{p}_{3}^{2}+{M}^{2}}}{{p}_{3}^{2}+{M}^{2}-{\mu }^{2}}{\rm{d}}{p}_{3},\end{eqnarray}$ where only the lowest Landau level contribution (n =0) is included. With the given model parameters, our numerical calculation shows that the region of ${eB}\gt 11{m}_{\pi }^{2}$ satisfies the so-called strong-field limit. Note that the magnitude ${eB}\approx 11{m}_{\pi }^{2}$ is larger than the threshold value, but approaches the estimated magnitude of the strong magnetic field in the peripheral collisions at RHIC (see, e.g. [31]).
Using the simplified criterion in equation (28 ), we find that BECI behaves as a monotonically increasing function for ${eB}\gt 11{m}_{\pi }^{2}$ . Moreover, this point has been reached in figure 5 (see the tendency for ${eB}\gt 11{m}_{\pi }^{2}$ ). This is the effect of magnetic catalysis, i.e. the emergence of BECI locates at a larger chemical potential with an increase in the strength of the magnetic field. In view of this, ${\mu }_{c}^{{\rm{I}}}$ determines the critical end point Q1, meaning that that the chiral transition is pushed towards a larger chemical potential. In this sense, the strong-field result shown here is consistent with the chiral magnetic catalysis predicted in the literature (see, e.g. [13 –15]).
In order to fully understand magnetic catalysis, we explore possible changes in the qualified model parameters which guarantee the emergence of BECI . For this purpose, we first focus on the effective diquark coupling parameter, $H^{\prime} $ . By using the simplified criterion from equation (28 ), its magnetic dependence is indicated by the solid line in figure 6 . We find that the qualified value of $H^{\prime} $ exhibits as a monotonically decreasing function of the magnetic field. This result actually originates from two different kinds of magnetic effects: the first can be ascribed to the Lowest Landau level contribution, as given in equation (28 ). The second takes effect implicitly through the magnetic dependence of constituent quark mass. If we were to neglect the latter, and assume an unchanged mass, the behavior of $H^{\prime} $ would be a more obvious decreasing function, as indicated by the dashed line in figure 6 . Either the solid line or the dashed line indicates that, with an increase in the magnetic field, a weaker strength of $H^{\prime} $ is required in order to achieve BECI transition.
Figure 6.
New window|Download| PPT slide Figure 6.Strong-field result of $H^{\prime} $ required for BECI transition. This is obtained at a given quark chemical potential, and $H^{\prime} $ is normalized with respect to its zero-field value ${H}_{0}^{{\prime} }$ .
Moreover, we consider the case of parameter $K^{\prime} $, which describes chiral-diquark interplay. While $H^{\prime} $ behaves as a decreasing function, it is observed that the value of $| \chi | $ increases with the magnetic field (i.e. the magnetic catalysis of the chiral condensate). Based on equation (13 ), where $H^{\prime} =H+\tfrac{1}{4}K^{\prime} | \chi | $, we draw the conclusion that $K^{\prime} $ displays a decreasing function in relation to the magnetic field. To summarize, the results of qualified $H^{\prime} $ and $K^{\prime} $ are clear in the strong-field limit. With an increased magnetic field, the weaker strength of $H^{\prime} $ and/or a smaller $K^{\prime} $ value, are required in order to achieve BECI transition. Physically speaking, this indicates that a strong magnetic background facilitates the emergence of BECI, and the coexistence of chiral and diquark condensates.
For finite magnetic fields, on the other hand, the above argument is no longer accessible. As shown in figure 5, the values of ${\mu }_{c}^{{\rm{I}}}$ decrease in the region of $6{m}_{\pi }^{2}\lt {eB}\lt 10{m}_{\pi }^{2}$ . For intermediate fields, the effect of inverse magnetic catalysis causes BECI transition. As Q1 plays the role of a critical end point, the decreasing function of ${\mu }_{c}^{{\rm{I}}}$ implies that the critical values of chiral transition decrease with an increasing magnetic field. Clearly, this contradicts the chiral magnetic catalysis observed in the strong-field limit. Indeed, such an inverse magnetic catalysis (or magnetic inhibition) has been observed in the two-color and two-flavor NJL model (without including axial anomaly) [28]. There, quarks with nonzero rotated charge and their coupling to the magnetic field were taken into account. The BEC phase considered in [28] is quite similar to the BECI phase shown here. Note, however, that the lattice calculations could not handle for three-color QCD at moderate density. In this sense, our result regarding inverse magnetic catalysis may prove to be more important, from the perspective of model-dependent studies.
5. Conclusions and outlook
Within an extended NJL model with axial anomaly, we investigated the influences of rotated magnetic fields on the coexistence of diquark and chiral condensates, together with related critical phenomena. As far as we know, this problem has not previously been addressed in the literature. The main result can be illustrated by means of a schematic phase diagram in the eB -μ plane.
For weak magnetic fields, it is shown in figure 7 that the coexistence region takes place in the way that BECI emerges first, followed by BEC. The difference from the vanishing magnetic field situation lies in the ‘splitting’ of the degenerated BEC phase. Correspondingly, the critical phenomena are basically similar to the zero-field result given in [7].
Figure 7.
New window|Download| PPT slide Figure 7.Schematic phase diagram in the eB -μ plane. The solid lines correspond to the second-order transitions, whereas the dotted line is a first-order transition.
A strong magnetic field significantly modifies the coexistence region predicted in the zero-field situation. As shown in figure 7, this region is superseded in favor of BECI, and the BEC phase disappears completely. This stems from the effect of magnetic catalysis for the BECI transition, i.e. a strong field leads to BECI emerging at a larger quark chemical potential. Moreover, due to magnetic catalysis, the qualified values of $H^{\prime} $ and $K^{\prime} $ for the BECI transition are suppressed (as illustrated in figure 6 ). Thus, the strong magnetic background field can be anticipated to facilitate the emergence of BECI .
On the other hand, a first-order transition occurs at the boundary between BECI and BCS. This implies that the previously predicted BEC–BCS crossover no longer exists. This result should be understood from the viewpoint of the magnetic effect for an usual color superconducting phase without chiral condensate. As is well known, a strong magnetic field can potentially break the color-flavor locked symmetry explicitly in the BCS phase [19, 20, 24]. Due to this magnetic effect, the symmetry pattern in the BCS phase is no longer consistent with that in the NG phase. As a result, the previously predicted critical phenomenon is eventually disrupted. In conclusion, it can be stated that the effects of a strong magnetic field must be analyzed from two different sides. On one side of the coin is the magnetic catalysis for the BECI transition, which facilitates the emergence of chiral-diquark coexistence. On the other side is the explicit breaking of color-flavor locked symmetry in the BCS phase, which obstructs the emergence of the BECI –BCS crossover.
In addition, the results presented here may shed light on the effect of inverse magnetic catalysis in three-color QCD. Where magnetic catalysis becomes dominant for strong fields, inverse catalysis makes sense for intermediate magnetic fields. Note that the regime of inverse catalysis ${eB}\sim 6-10{m}_{\pi }^{2}\sim {10}^{18-19}{\rm{G}}$ approaches the estimated magnitude of the magnetic field inside magnetars [32, 33]. On the assumption that CFL matter constitutes the interiors of stellar objects, the BECI phase and the inverse magnetic catalysis could therefore have potential relevance for the physics of magnetars. Possible applications remain for future publications.
In this paper, we consider three-flavor quark matter where the current quark mass mq =0 (the chiral limit). This involves quasiquarks with a rotated electric charge $\widetilde{Q}$, defined in color-flavor space, and their coupling to the magnetic field. The advantage of this treatment is that only the magnetic-induced splitting of the diquark condensate is taken into account. Owing to the influence of the magnetic field, the constituent quark masses are observed to be much larger than the current masses (including ms ). In this sense, it is feasible to adopt the degenerated constituent mass, as we have done in this work. If the flavor asymmetry (in particular ${m}_{s}\ne {m}_{u,d}$ ) were considered, the splitting originated from it (such as ${s}_{22}\ne {s}_{\mathrm{55,77}}$ ) would also require to be taken into account. This topic has previously been studied in the absence of magnetic fields. In [8], for instance, the two-flavor color superconducting phase, the CFL phase, and their respective BECs were included, so that the phase diagram became complicated. In an extended model with axial anomaly, it is rather difficult to explore both the magnetic effect and the effect of flavor asymmetry simultaneously, since (at least) four order parameters are involved.
In fact, the results shown in figure 7 represent one possible phase structure of CFL quark matter in the presence of magnetic fields. To date, the moderate density phase diagram in realistic situations remains an open problem. In addition to the scenario considered in this work, an alternative perspective is that spatial inhomogeneous phases appear in the region between the NG phase and the color superconducting phase [34 –36]. This represents a further opportunity for exploring the QCD phase structure. This falls beyond the scope of the present work, and investigations along these line are currently a work in progress.
Acknowledgments
The authors thank Hui Wang for the technical assistance. This work is supported by the National Natural Science Foundation of China (NSFC) under Contract No. 10875058.
AlfordM GRajagopalKWilczekF1998 QCD at finite baryon density: nucleon droplets and color superconductivity Phys. Lett. B 422 247256 247–56 DOI:10.1016/S0370-2693(98)00051-3 [Cited within: 4]
YamamotoNTachibanaMHatsudaTBaymG2007 Phase structure, collective modes, and the axial anomaly in dense QCD Phys. Rev. D 76 074001 DOI:10.1103/PhysRevD.76.074001 [Cited within: 3]
AbukiHBaymGHatsudaTYamamotoN2010 Nambu–Jona–Lasinio model of dense three-flavor matter with axial anomaly: the low temperature critical point and BEC–BCS diquark crossover Phys. Rev. D 81 125010 DOI:10.1103/PhysRevD.81.125010 [Cited within: 11]
BaslerHBuballaM2010 Role of two-flavor color superconductor pairing in a three-flavor Nambu–Jona–Lasinio model with axial anomaly Phys. Rev. D 82 094004 DOI:10.1103/PhysRevD.82.094004 [Cited within: 2]
PowellP DBaymG2012 Axial anomaly and the three-flavor Nambu–Jona–Lasinio model with confinement: Constructing the QCD phase diagram Phys. Rev. D 85 074003 DOI:10.1103/PhysRevD.85.074003
PowellP DBaymG2013 Asymmetric pairing of realistic mass quarks and color neutrality in the Polyakov–Nambu–Jona–Lasinio model of QCD Phys. Rev. D 88 014012 DOI:10.1103/PhysRevD.88.014012 [Cited within: 1]
GusyninV PMiranskyV AShovkovyI A1995 Dimensional reduction and dynamical chiral symmetry breaking by a magnetic field in 3 + 1 dimensions Phys. Lett. B 349 477483 477–83 DOI:10.1016/0370-2693(95)00232-A [Cited within: 3]
MiranskyV A1996 Catalysis of dynamical symmetry breaking by a magnetic field Prog. Theor. Phys. Suppl. 123 4960 49–60 DOI:10.1143/PTPS.123.49
GusyninV PMiranskyV AShovkovyI A1996 Dimensional reduction and catalysis of dynamical symmetry breaking by a magnetic field Nucl. Phys. B 462 249290 249–90 DOI:10.1016/0550-3213(96)00021-1 [Cited within: 3]
RenC-FZhangX-BZhangY2014 Magnetic effects in color-flavor locked superconducting phase with the additional chiral condensates Chin. Phys. Lett. 31 062501 DOI:10.1088/0256-307X/31/6/062501 [Cited within: 1]
BuballaMOertelM2002 Color-flavor unlocking and phase diagram with self-consistently determined strange-quark masses Nucl. Phys. A 703 770784 770–84 DOI:10.1016/S0375-9474(01)01674-8 [Cited within: 3]
EbertDKlimenkoK GVdovichenkoM AVshivtsevA S1999 Magnetic oscillations in dense cold quark matter with four-fermion interactions Phys. Rev. D 61 025005 DOI:10.1103/PhysRevD.61.025005 [Cited within: 1]
FrascaMRuggieriM2011 Magnetic susceptibility of the quark condensate and polarization from chiral models Phys. Rev. D 83 094024 DOI:10.1103/PhysRevD.83.094024 [Cited within: 1]
DuarteD CAllenP GFariasR L SMansoP H ARamosR OScoccolaN N2016 BEC–BCS crossover in a cold and magnetized two color NJL model Phys. Rev. D 93 025017 DOI:10.1103/PhysRevD.93.025017 [Cited within: 3]
KitazawaMRischkeD HShovkovyI A2008 Bound diquarks and their Bose–Einstein condensation in strongly coupled quark matter Phys. Lett. B 663 228233 228–33 DOI:10.1016/j.physletb.2008.03.067 [Cited within: 1]
SkokovVIllarionovA YToneevV2009 Estimate of the magnetic field strength in heavy-ion collisions Int. J. Mod. Phys. A 24 59255932 5925–32 DOI:10.1142/S0217751X09047570 [Cited within: 1]
IbrahimA I et al. 2004 Discovery of a transient magnetar: Xte j1810-197 Astrophys. J. Lett. 609 L21 DOI:10.1086/422636 [Cited within: 1]
FerrerE Jde la InceraVKeithJ PPortilloISpringsteenP L2010 equation of state of a dense and magnetized fermion system Phys. Rev. C 82 065802 DOI:10.1103/PhysRevC.82.065802 [Cited within: 1]
HeinzAGiacosaFWagnerMRischkeD H2016 Inhomogeneous condensation in effective models for qcd using the finite-mode approach Phys. Rev. D 93 014007 DOI:10.1103/PhysRevD.93.014007
BraunJKarbsteinFRechenbergerSRoscherD2016 Crystalline ground states in Polyakov-loop extended Nambu–Jona–Lasinio models Phys. Rev. D 93 014032 DOI:10.1103/PhysRevD.93.014032 [Cited within: 1]
,1, Yun-Ben Wu(吴云奔)1, Yi Zhang(张一)21 School of Physics, 
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
New window|Download| PPT slide
