Phase diagram of two-flavor quark matter: Gluonic phase at nonzero temperature
aa r X i v : . [ h e p - ph ] J a n Phase diagram of two-flavor quark matter: Gluonic phase at nonzero temperature
O. Kiriyama ∗ Institut f¨ur Theoretische Physik, J.W. Goethe-Universit¨at, D-60438 Frankfurt am Main, Germanyand Research Center for Nuclear Physics, Osaka University, Ibaraki 567-0047, Japan (Dated: November 7, 2018)The phase structure of neutral two-flavor quark matter at nonzero temperature is studied. Ouranalysis is performed within the framework of a gauged Nambu–Jona-Lasinio model and the mean-field approximation. We compute the free energy of the gluonic phase (gluonic cylindrical phaseII) in a self-consistent manner and investigate the phase transition from the gluonic phase to the2SC/g2SC/NQ phases. We briefly consider the phase diagram in the plane of coupling strengthversus temperature and discuss the mixed phase consisting of the normal quark and 2SC phases.
PACS numbers: 12.38.-t, 11.30.Qc, 26.60.+c
I. INTRODUCTION
The properties of cold and dense quark matter are of great interest in astrophysics and cosmology. In particular,at moderate densities of relevance for the interior of compact stars, quark matter is a color superconductor and hasa rich phase structure with important implications for compact star physics [1, 2, 3, 4, 5, 6, 7, 8, 9].Bulk matter in the interior of compact stars should be color and electrically neutral and be in β -equilibrium. Inthe two-flavor case, these conditions separate the Fermi momenta of up and down quarks and, as a consequence,the ordinary BCS state (2SC) is not always energetically favored over other unconventional states. The possibilitiesinclude crystalline color superconductivity and gapless color superconductivity (g2SC) [10, 11, 12, 13]. However,the 2SC/g2SC phases suffer from a chromomagnetic instability, indicated by imaginary Meissner masses of somegluons [14, 15]. The instability related to gluons of color 4–7 occurs when the ratio of the gap over the chemicalpotential mismatch, ∆ /δµ , decreases below a value √
2. Resolving the chromomagnetic instability and clarifying
50 100 150 2000204060 ∆∆∆∆ [MeV] T [ M e V ] NQ 2SCg2SC µ = 400 MeV FIG. 1: The phase diagram of electrically neutral two-flavor quark matter in the plane of ∆ and T . At T = 0, the g2SCphase exists in the window 92 MeV < ∆ <
134 MeV and the 2SC window is given by ∆ >
134 MeV. The unstableregion for gluons 4–7 is depicted by the region enclosed by the thick solid line. The g2SC phase and a part of the 2SC phase(92 MeV < ∆ <
162 MeV) suffer from the chromomagnetic instability at T = 0. The quark chemical potential is taken to be µ = 400 MeV. ∗ Electronic address: [email protected] the nature of true ground state of dense quark matter are central issues in the study of color superconductivity[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. (For a three-flavor case, see Refs.[38, 39, 40, 41, 42, 43, 44].)As an example, in Fig. 1, we plot the phase diagram of neutral two-flavor quark matter in the plane of the 2SCgap at δµ = 0 (∆ ) and temperature ( T ) [31, 32]. (The parameter ∆ is essentially the diquark coupling strength.)In order to obtain this diagram, we employed a gauged Nambu–Jona-Lasinio (NJL) model, which is the very samemodel that we shall use in this paper. Furthermore, we neglected the color chemical potential. The 2SC/g2SC phasesand the unpaired normal quark (NQ) phase were included in the analysis. The quark chemical potential was takento be µ = 400 MeV, which is a value typical for the cores of compact stars. The region enclosed by the thick solidline is unstable (because gluons of color 4–7 have tachyonic Meissner masses there) and, therefore, should be replacedby other chromomagnetically stable phases, for instance, gluonic phases [24, 25]. (For more detailed discussions ofthe gluonic phases, see Refs. [45, 46].) Note, however, that we did not consider the global structure of a free energyin extracting the unstable region, but only the tendency toward the vector condensation h ~A i in the 2SC/g2SCphases. A self-consistent analysis of the gluonic phases at T = 0 has recently been done by Hashimoto and Miransky[36] and they found that the gluonic phase (strictly speaking, the gluonic cylindrical phase II) exists in the window65 MeV < ∆ <
160 MeV and is energetically more favored than the 2SC/g2SC/NQ phases in this whole window.In this paper, we study the gluonic cylindrical phase II at nonzero temperature and revisit the phase diagram shownin Fig. 1, computing the free energy of the gluonic phase in a self-consistent manner. The result would be useful forthe phase diagram of QCD, and the compact star phenomenology as well.
II. MODEL
In order to study the gluonic phase, we use the gauged NJL model with massless up and down quarks: L = ¯ ψ ( iD/ + ˆ µγ ) ψ + G D (cid:0) ¯ ψiγ εǫ b C ¯ ψ T (cid:1) (cid:0) ψCiγ εǫ b ψ (cid:1) − F aµν F aµν , (1)where the quark field ψ carries flavor ( i, j = 1 , . . . N f with N f = 2) and color ( α, β = 1 , . . . N c with N c = 3) indices, C is the charge conjugation matrix; ( ε ) ik = ε ik and ( ǫ b ) αβ = ǫ bαβ are the antisymmetric tensors in flavor and colorspaces, respectively. The diquark coupling strength in the scalar ( J P = 0 + ) color-antitriplet channel is denoted by G D . The covariant derivative and the field strength tensor are defined as D µ = ∂ µ − igA aµ T a , (2a) F aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν . (2b)To evaluate loop diagrams we use a three-momentum cutoff Λ = 653 . µ and µ e ) by hand[47] to ensure color- and electric-charge neutrality. In β -equilibrated neutral two-flavor quark matter, the elements ofthe diagonal matrix of quark chemical potentials ˆ µ are given by µ ur = µ ug = ¯ µ − δµ,µ dr = µ dg = ¯ µ + δµ,µ ub = ¯ µ − δµ − µ ,µ db = ¯ µ + δµ − µ , (3)with ¯ µ = µ − δµ µ , δµ = µ e . (4)In Nambu-Gor’kov space, the inverse full quark propagator S − ( p ) is written as S − ( p ) = (cid:18) ( S +0 ) − Φ − Φ + ( S − ) − (cid:19) , (5)with ( S +0 ) − = γ µ p µ + (¯ µ − δµτ ) γ + gγ µ A aµ T a , (6a)( S − ) − = γ µ p µ − (¯ µ − δµτ ) γ − gγ µ A aµ T aT , (6b)and Φ − = − iεǫ b γ ∆ , Φ + = − iεǫ b γ ∆ . (7)Here τ = diag(1 , −
1) is a matrix in flavor space. Following the usual convention, we have chosen the diquarkcondensate to point in the third direction in color space.For the gluonic cylindrical phase II, B = h gA z i is the most relevant condensate, because the chromomagneticinstability related to gluons 4–7 corresponds to the tachyonic mode in the direction of B .[24, 25, 30] Besides B , wehave to introduce a color chemical potential µ = h gA i to ensure color neutrality at B = 0. Taking into accountthese condensates, the free energy of the gluonic phase in the one-loop approximation is given by Ω (∆ , µ e , µ , B, µ ; µ, T )= − µ B g + µ µ B g − µ B g − π (cid:18) µ e + 2 π T µ e + 7 π T (cid:19) + ∆ G D − X a Z d p (2 π ) h | ǫ a | + 2 T ln(1 + e − β | ǫ a | ) i , (8)where β = 1 /T , the ǫ a ’s are quasi-quark energies and the sum runs over all particle and anti-particle ǫ a ’s. Here, weadded tree-level contributions from gluons (first line on the r.h.s.), Ω (tree) g = g f abc f ade A bµ A cν A dµ A eν , (9)and electrons (second line on the r.h.s.). Note also that the ǫ a ’s depend on the vector condensates through thecovariant derivatives in the quark propagator (5). In what follows, we neglect the color chemical potentials µ , and,consequently, the tree-level contributions of gluons. We have carefully checked that their effect on the free energy isnegligible for realistic values of α s ≃ Ω R = Ω (∆ , µ e , B ; µ, T ) − Ω (0 , , B ; 0 , . (10)It is known that this free energy subtraction is not adequate to remove the cutoff dependence of the free energy at T >
0. In fact, Eq. (10) leads to positive Meissner screening masses in the normal phase at
T > µ = 400 MeV and at the temperatures of interest (20 MeV at most).In order to find the neutral gluonic phase, we first solve a set of coupled equations (the gap equation and theelectrical charge neutrality condition), ∂Ω R ∂ ∆ = ∂Ω R ∂µ e = 0 , (11)as a function of B and, then, compute the free energy Ω R ( B ). Finally, the minimum of Ω R ( B ) determines the neutralgluonic phase. (In the following Figs. 3 and 4, we plot the free energy evaluated along the solution of the coupledequations (11).) III. NUMERICAL RESULTS
Figure 2 shows ∆, δµ and B in the gluonic phase at T = 0 as a function of ∆ . First, let us note that the results ofFig. 2 are in good agreement with those shown in Figs. 1, 2, and 3 of Ref. [36], where the color chemical potentials µ , were treated self-consistently. In Fig. 2 one can see that the gluonic phase exists in the window66 MeV < ∆ <
162 MeV . (12)The gluonic phase is energetically favored over the 2SC/g2SC/NQ phases in this whole window (see also Fig. 5 of Ref.[36]). One also sees that the phase transition between the gluonic phase and the NQ (2SC) phase at ∆ = 66 MeV(162 MeV) is strongly (weakly) of first order.
50 100 1500100200300 B δµ ∆∆∆∆∆ [MeV] [ M e V ] FIG. 2: The gap parameter ∆ (solid line), the chemical potential mismatch δµ (dotted line) and the gluonic vector condensate B (dashed line) versus ∆ in the gluonic phase at T = 0. The quark chemical potential is taken to be µ = 400 MeV. B [MeV] F ree E n er gy [ M e V / f m ] ∆ =75 MeV ∆ =85 MeV ∆ =100 MeV ∆ =140 MeV FIG. 3: The free energy Ω R ( B ) as a function of B at T = 0 for ∆ = 75 MeV (solid line), ∆ = 85 MeV (dotted line),∆ = 100 MeV (dot-dashed line), and ∆ = 140 MeV (dashed line). Note that the free energy is measured with respect to the2SC/g2SC/NQ phases at B = 0. The results are plotted for µ = 400 MeV. Now let us take a closer look at the free energy at T = 0. Figure 3 shows the behavior of Ω R ( B ) measured withrespect to the 2SC/g2SC/NQ phases at B = 0. The results are plotted for µ = 400 MeV at several values of ∆ .In the weak coupling regime, 66 MeV < ∆ <
92 MeV, the chromomagnetic instability does not exist in the NQphase (see Fig. 1). We note that the curvature of Ω R ( B ) at B = 0, m M = d Ω R ( B ) dB (cid:12)(cid:12)(cid:12)(cid:12) B =0 , (13)can be regarded as the Meissner mass squared ∂ Ω R /∂B | B =0 in the 2SC/g2SC/NQ phases, since the solutions ofEq. (11) satisfy ∆ = ¯∆ + O ( B ) and µ e = ¯ µ e + O ( B ) for small values of B , where ¯∆ and ¯ µ e denote their valuesat B = 0 [24, 25]. We found that m M is indeed zero in the weak coupling regime. In addition, we observed that,for small values of B , the system is in the ungapped (∆ = 0) phase and the free energy behaves like Ω R ∼ O ( B ). B [MeV] B [MeV] B [MeV] B [MeV] F ree E n er gy [ M e V / f m ] F ree E n er gy [ M e V / f m ] F ree E n er gy [ M e V / f m ] F ree E n er gy [ M e V / f m ] T =0 T =4 T =8 T =14 T =0 T =9 T =18 T =24 T =0 T =10 T =18 T =24 T =0 T =6 T =12 T =18 (a) (b)(c) (d) FIG. 4: The temperature dependence of the free energy (measured with respect to the 2SC/g2SC/NQ phases at B = 0) as afunction of B for ∆ = 75 MeV (a), for ∆ = 85 MeV (b), for ∆ = 100 MeV (c), and for ∆ = 140 MeV (d). The results areplotted for µ = 400 MeV and the values of T are given in MeV. However, contrary to the result of Fig. 1, the free energy has a global minimum at B = 0 and the gluonic phaseis energetically favored over the NQ phase. For 92 MeV < ∆ .
162 MeV, one finds tachyonic modes at B = 0because the g2SC phase and a part of the 2SC phase suffer from the chromomagnetic instability and, therefore, areunstable against the formation of B . Consequently, the gluonic phase is realized in this region, as expected. Forstrong coupling, ∆ &
162 MeV, the 2SC phase is chromomagnetically stable in this regime and the free energy hasa global minimum at B = 0, though it is not plotted in Fig. 3.We now turn to the free energy of the gluonic phase at T >
0. Figure 4(a) display the temperature dependenceof the free energy for ∆ = 75 MeV. As T grows, the free-energy gain gets reduced, but the change of the vacuumexpectation value of B is rather small. As a result, we observe a strong first-order transition from the gluonic phaseto the NQ phase at T ≃
14 MeV. Note that m M remains positive at any value of T , which is consistent with theresult shown in Fig. 1.In Fig. 4(b), the same plot is displayed for ∆ = 85 MeV. At low temperature, like in the case of ∆ = 75 MeV,the gluonic phase is more favored than the chromomagnetically stable NQ phase. At T ≃ m M turns negative,meaning that the stable NQ phase undergoes a phase transition into the unstable g2SC phase (see Fig. 1). Thegluonic phase is energetically favored until the temperature reaches T ≃
20 MeV. Above this temperature, the g2SCphase becomes stable and therefore is favored.In Figs. 4(c) and 4(d), we plot the free energy for the cases of ∆ = 100 MeV and ∆ = 140 MeV, respectively.In both cases, the gluonic phase is energetically favored at low temperature. In contrast, at high temperature, theglobal minima of the free energy are realized at B = 0, which means that, as expected from the result of Fig. 1, the
50 100 150 2000204060 weak 1st order1st order 1st order ∆∆∆∆ [MeV] T [ M e V ] NQ 2SCg2SC gluonic cyl. II
FIG. 5: Schematic phase diagram of neutral two-flavor quark matter at moderate density in ∆ - T plane. The thick solid linedenotes the line of second-order or weakly first-order transitions and strong first-order transitions are indicated by a thickdashed line. In the region enclosed by the thick solid and dashed lines, the gluonic phase is energetically more favored thanthe 2SC/g2SC/NQ phases. chromomagnetically stable 2SC/g2SC phases are favored. For ∆ = 100 MeV, the phase transition from the gluonicphase to the g2SC phase takes place at T ≃
21 MeV. In the case of ∆ = 140 MeV, the phase transition takes placeat T ≃
18 MeV.Here, we would like to make a comment regarding the order of the phase transitions. As mentioned above, weobserved the strong first-order transition (gluonic phase ↔ NQ phase) at ∆ = 75 MeV. On the other hand, for thecases of ∆ = 85 , ,
140 MeV, the phase transition (gluonic phase ↔ IV. SUMMARY, CONCLUSIONS, AND OUTLOOKA. Phase diagram
We studied the gluonic cylindrical phase II at nonzero temperature. Using the gauged NJL model and the one-loopapproximation, we computed the free energy of the gluonic phase self-consistently and investigated the phase structureof the gluonic phase. Although we neglected the color chemical potentials, we have checked that, for α s ≃
1, theireffect on the free energy is negligible.In the weak coupling regime, we found that the gluonic phase undergoes a strong first-order transition into the NQphase as it is heated. This is a new aspect of the gluonic phase at
T >
0, which is not shown in Fig. 1. On theother hand, since the phase transitions from the gluonic phase to the chromomagnetically stable 2SC/g2SC phasesare of second-order or weakly first-order, we expect that the corresponding critical line shown in Fig. 1 (i.e., the rightbranch of the thick solid line) is not drastically altered by the self-consistent analysis. (In other words, the Meissnermasses squared can be a rough criterion for choosing the energetically favored phase in this regime.) We thus are ableto make a sketch of a schematic phase diagram of two-flavor quark matter, which is free from the chromomagneticinstability related to gluons 4–7 (see Fig. 5). The low-temperature region of the g2SC phase and a part of the2SC phase is replaced by the gluonic phase. Furthermore, the gluonic phase wins against a part of the NQ phase and It is interesting to note that the neutral single plane-wave Larkin-Ovchinnikov-Fulde-Ferrell state [48, 49] has a similar phase structureas the gluonic phase [26, 31, 32, 34].
100 150−20−100 100 150−20−100 F ree E n er gy [ M e V / f m ] ∆∆∆∆ [MeV] ∆∆∆∆ [MeV] σ =3 MeV/fm σ =10 MeV/fm NQ NQ
FIG. 6: The free energy of the neutral 2SC/g2SC phase (solid line), the gluonic phase (dotted line), and the mixed phase(dashed line) measured with respect to the NQ phase as a function of ∆ for σ = 3 MeV / fm (left) and σ = 10 MeV / fm (right). The three dots on the solid line (∆ = 92 , ,
162 MeV from left to right) denote the edge of the g2SC window with thenormal phase, the phase transition point between the 2SC and the g2SC phases, and the critical point of the chromomagneticinstability. The quark chemical potential is taken to be µ = 400 MeV. enlarges its region. We argue therefore that the gluonic phase which could resolve the chromomagnetic instabilityrelated to gluons 4–7 is a strong candidate for the ground state of a neutral two-flavor color superconductor in theintermediate coupling regime. Alternatives include other types of the gluonic phases [24, 25, 36, 37], the crystallinephases [10, 11, 43] and the mixed phase [19]. It should be mentioned that, at T = 0, the gluonic color-spin locked phaseis more stable than the gluonic cylindrical phase II in some region of ∆ and moreover is free from the chromomagneticinstability at moderate densities [54, 55].Although we concentrated on the phase diagram in T -∆ plane in this work, it is obviously worthwhile to revisitthe phase diagram in T - µ plane. A preliminary study [50] indicates that currently known phase diagrams [51, 52, 53]must be significantly altered. In addition, the critical temperature for the gluonic phase could reach a few tens ofMeV and, therefore, it is interesting to study astrophysical implications of the gluonic phase, e.g., the quark matterequation of state, neutrino emission from compact star cores, and so on. B. Gluonic phase versus mixed phase
Finally we briefly look at a mixed phase consisting of the NQ and the 2SC phases [19, 20, 21]. For the mixed phaseto exist, it must satisfy the Gibbs conditions, which are equivalent to chemical and mechanical equilibrium conditionsbetween the NQ and the 2SC phases. These conditions end up as follows P (NQ) ( µ, µ e ) = P (2SC) ( µ, µ e ) . (14)Beside Eq. (14) two components must have opposite electrical charge densities. Otherwise a globally neutral mixedphase could not exist. We solved Eq. (14) and found that the globally neutral mixed phase exists in the window67 MeV < ∆ <
201 MeV , (15)where the quark chemical potential was taken to be µ = 400 MeV.In order to calculate the free energy of the mixed phase we take account of finite-size effects, i.e., the surface andCoulomb energies associated with phase separation. The surface and Coulomb energy densities are given by ǫ S = dxσr , ǫ C = 2 πα em f d ( x ) x ( ∆n e ) r , (16)where σ is the surface tension, x is the volume fraction of the rarer phase, ∆n e is the difference of the electric chargedensity between NQ and 2SC phases, and α em = 1 / d ( d = 1 ,
2, and 3 correspond to slabs, rods, and droplets configurations, respectively) and r , which denotes the radiusof the rarer phase. The geometrical factor f d ( x ) is given by f d ( x ) = 1 d + 2 (cid:18) − dx − /d d − x (cid:19) . (17)Minimizing the sum of ǫ S and ǫ C with respect to r , we obtain ǫ S + ǫ C = 32 (cid:0) πα em d f d ( x ) x (cid:1) / ( ∆n e ) / σ / . (18)The actual value of the surface tension in quark matter is poorly known, in this work we assume d = 3 (dropletsconfiguration) and try relatively small surface tension.Figure 6 displays the free energy of the 2SC/g2SC phase, the gluonic phase, and the mixed phase. For a very smallsurface tension σ = 3 MeV / fm , the mixed phase is the most favored in a wide range of ∆ , 103 MeV < ∆ <
166 MeV.The gluonic phase is energetically more favored than the mixed phase only in the weak coupling regime. Note thatthe value of the surface tension, σ = 3 MeV / fm at µ = 400 MeV, is close to that calculated by Reddy and Rupak[19]. For a surface tension σ = 10 MeV / fm , there still is a wide window where the mixed phase is more stable thanthe g2SC phase, but the mixed phase is less favored than the gluonic phase.It should be mentioned here that, however, we did not take into account the thickness of the boundary layer, whichhas been estimated to be comparable to the value of the Debye screening length in each of the two phases, andtherefore the results shown in Fig. 6 is not a final conclusion [56, 57]. The effect of charge screening would increasethe surface energy substantially [58, 59]. Acknowledgments
I would like to thank Dirk Rischke and Armin Sedrakian for discussions and for comments on the earlier versionof the manuscript. I also would like to thank H. Abuki, M. Ruggieri, and I. Shovkovy for discussions during theYITP international symposium “Fundamental Problems in Hot and/or Dense QCD”. This work was supported bythe Deutsche Forschungsgemeinschaft (DFG). [1] K. Rajagopal and F. Wilczek, in