A more realistic holographic model of color superconductivity with the higher derivative corrections
AA more realistic holographic model of color superconductivity inEinstein-Gauss-Bonnet gravity
Cao H. Nam ∗ Phenikaa Institute for Advanced Study and Faculty of Fundamental Sciences,Phenikaa University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam (Dated: January 5, 2021)In this paper, we have constructed a bottom-up holographic model for the color supercon-ductivity (CSC) of the Yang-Mills theory in the context of Einstein-Gauss-Bonnet (EGB)gravity, which allows to study the CSC phase with the color number N c ≥
2. We analyze theCooper pair condensate in the deconfinement and confinement phases which are dual to theplanar GB-RN-AdS black hole and GB-AdS soliton, respectively, with including the back-reaction of the matter part. By examining the breakdown of the Breitenlohner-Freedmanbound in the background of the planar GB-RN-AdS black hole, we find that the upper boundof the color number with the GB coupling parameter α > N c ≥ α < N c ≥ α . This is confirmed and the corresponding phase diagram is found by solving numericallythe equations of motion for the gravitational system. In addition, we show that the CSCphase disappears in the confinement phase for the amplitude of α below a certain value whichmeans that beyond that value it may lead to the breakdown region of the EGB gravity ininvestigating the CSC phase. I. INTRODUCTION
It is expected in quantum chromodynamics (QCD) that at sufficiently high chemical potential(density) and low temperature quarks condense into Cooper pairs in analogy to the condensation ofelectrons in the conventional metallic superconductors [1]. Unlike the condensation of the electronpairs where the Coulomb interaction between them is repulsive and has to be overcome by an at-traction caused by the coupling between electrons with phonons, the strong interaction between twoquarks is attractive (in the color-antisymmetric channel) and thus the Bardeen-Cooper-Schrieffer(BCS) mechanism applied to the quark pairs is more direct than its original setting. The quark ∗ Electronic address: [email protected] a r X i v : . [ h e p - t h ] J a n pairs carry the net color charge or in other words they are gauge non-invariant operators. There-fore, the condensation of the quark pairs breaks spontaneously the SU (3) C gauge symmetry ofQCD and gives rise the masses for the gluon via Higgs mechanism. This phenomena is thus re-ferred to the color superconductivity (CSC). It is interesting to study the CSC phase from boththeoretical and phenomenological aspects. The quark pairs have color and flavor degrees of freedombesides the spin one and hence there are different condensation patterns of which the color-flavorlocked phase [4] is well-known. Also, the CSC phase might occur in the cores of neutron stars withthe densities possibly reaching up ten times nuclear-matter saturation density.At very large temperature or chemical potential, QCD becomes weakly coupled due to theasymptotic freedom and hence an exact analytic study of the quark matter is possible. However,in the nonperturbative region the investigations are mainly based the phenomenological models atwhich many important features are missed. Many nonperturbative investigations can be performedby using the numerical simulation, however it can be inaccessible at finite chemical potential dueto the sign problem of the Euclidean action.Another approach for investigating the properties of the strongly coupled theories at finite tem-perature and chemical potential is via the AdS/CFT correspondence [5–7] which relates the weaklycoupled gravitational theory in d -dimensional AdS spacetime and the strongly coupled conformaltheory of d − U (1) gauge field anda real scalar field in six dimensions with the boundary geometry R , × S . The scalar field tendsto condense in the near-horizon region at a very low temperature and the CSC phase transitionis found due to the fact that the near-horizon geometry of planar Reissner–Nordstrom (RN) AdSblack hole is AdS × R [29, 33] corresponding to the new instability bound. However, the real It was indicated that the gauge non-invariant operators constructed by the quark pairs are suppressed in the limitof large color number [2, 3], which is considered as one of the large obstacles for the relevant investigations. scalar field does not correspond to a diquark operator. Therfore, Ref. [31] considered the complexscalar field (rather than the real one) whose U (1) charge is regarded as the baryon number ofthe diquark operator. In this work, the backreaction of the matter fields is ignored and the CSCphase is found to appear above a critical chemical potential. In particular, a detail and profoundinvestigation about the CSC phase transition in the Yang-Mills (YM) theory is performed in Ref.[32] where the authors investigated the CSC phase transition for both the deconfinement and theconfinement phases with including the backreation of the matter part. The authors indicated thatthere is the CSC phase transition in the decconfinement phase but not in the confinement phasefor the color number N c = 1. But, for N c ≥ N c ≥
2. In order to do that, we instead construct a gravitational dual modelin Einstein-Gauss-Bonnet (EGB) gravity which is an extension of Einstein gravity with includingthe higher curvature corrections written as the Gauss-Bonnet (GB) term. We analyze the CSCphase transition in both the decconfinement and the confinement phases which are dual to theplanar GB-RN-AdS black hole and GB-AdS soliton, respectively, and indicate the role of the GBterm on the occurrence of the CSC phase for N c ≥ N c ≥ II. MODEL SETUP
In this section, we introduce the gravitational dual model in the framework of the six-dimensional EGB gravity for the CSC phase transition, given by the following action S bulk = 12 κ (cid:90) d x √− g [ R −
2Λ + (cid:101) α L GB + L mat ] , (1)where Λ is the cosmological constant defined in terms of the asymptotic AdS radius l as Λ = − l , L GB is the GB term given by L GB = R − R µν R µν + R µνρλ R µνρλ , (2) (cid:101) α is the GB coupling parameter, and L mat is the matter Lagrangian. In the bottom-up construction,the matter Lagrangian for the holographic model consists of a U (1) gauge field A µ and a complexscalar field ψ as L mat = − F µν F µν − | ( ∇ µ − iqA µ ) ψ | − m | ψ | . (3)In this Lagrangian, the U (1) gauge field is regarded as the dual description of the current of thebaryon number whose time component will describe the baryon charge density and the chemicalpotential of the quarks, whereas the complex scalar field ψ is dual to the diquark operator inthe boundary field theory and q is its U (1) charge which is regarded as the baryon number ofthe diquark operator. Note that, the baryon number of the diquark operator is related the colornumber N c as q = N c . In the following, we set 1 / κ = 1 and l = 1.Note that, the GB term can be naturally obtained from the low-energy limit of heterotic stringtheory [34–38] where the GB coupling parameter (cid:101) α is regarded as the inverse string tension andthus (cid:101) α >
0. However, in the bottom-up model we consider either (cid:101) α > (cid:101) α <
0. It was pointedout in [39, 40], there are the constraints imposed by the causality of the boundary field theory as, − / ≤ α ≤ / α ≡ (cid:101) α . As indicated later by Hofman [41], the bounds obtainedfrom the causality constraints of the boundary field theory should not be a feature of the thermalCFTs but the causality violation reflects a fact that the interaction can occur in the asymptoticregion close to the boundary. Also, for the better understanding of the effects of the GB term onthe CSC phase transition, in this work we permit the following range of the GB coupling parameter, α ∈ ( −∞ , / G µν + αH µν − l = T µν , ∇ ν F µν + iq [ ψ ∗ ( ∇ µ − iqA µ ) ψ − ψ ( ∇ µ + iqA µ ) ψ ∗ ] = 0 , ( ∇ µ − iqA µ )( ∇ µ − iqA µ ) ψ − m ψ = 0 , (4)where H µν = 2 (cid:16) RR µν − R µσ R σν − R µσνρ R σρ + R ρσλµ R νρσλ (cid:17) − g µν L GB ,T µν = 12 F µλ F ν λ + 12 [( ∇ ν − iqA ν ) ψ ( ∇ µ + iqA µ ) ψ ∗ + µ ↔ ν ] + 12 g µν L mat . (5)In order to solve these equations of motion, we first need to take the ansatz for the metric, vector,and scalar fields. We are interested in two solutions of the first equation of Eq. (4) which are dualto the deconfinement and confinement phases in the boundary field theory. More specifically, theansatz for the metric field is given by Eqs. (7) and (17) corresponding to the deconfinement andconfinement phases, respectively. The ansatz for the vector and scalar fields read A µ dx µ = φ ( r ) dt, ψ = ψ ( r ) . (6)For each of the deconfinement and confinement phases, we study the CSC phase transition bysolving Eq. (4) with the suitable boundary conditions in order to find the configuration with orwithout nontrivial scalar which the value of the scalar field is nonzero.The CSC phase appears due to the condensation of the scalar field corresponding to the spon-taneously broken U (1) symmetry. In the canonical ensemble where the charge is kept fixed, thecondensation of the scalar field is triggered by the chemical potential associated with the quarknumber density. Near the critical chemical potential, the value of the scalar field approaches zeroand since the backreaction of the scalar field on the spacetime metric is negligible. On the otherhand, the backreaction of the matter on the spacetime metric in this situation only comes fromthe vector field.The spacetime geometry dual to the deconfinement phase is the planar black hole solution whoseline element is given by the following ansatz ds = r (cid:0) − f ( r ) dt + h ij dx i dx j + dy (cid:1) + dr r f ( r ) , (7)where h ij dx i dx j = dx + dx + dx is the line element of the 3-dimensional planar hypersurface,and the direction y is compacted with the circle radius R y . The event horizon radius r + satisfies f ( r + ) = 0. The temperature of the boundary field theory is identified as the Hawking temperatureas, T = r f (cid:48) ( r + )4 π . In this configuration of the spacetime geometry, we find the equations for f ( r ), φ ( r ), and ψ ( r ) from Eq. (4) as α (cid:2) f (cid:48) ( r ) r + 3 f ( r ) (cid:3) f ( r ) − rf (cid:48) ( r ) − f ( r ) + 5 l = 18 [ φ (cid:48) ( r )] , (8) φ (cid:48)(cid:48) ( r ) + 4 r φ (cid:48) ( r ) − q ψ ( r ) r f ( r ) φ ( r ) = 0 , (9) ψ (cid:48)(cid:48) ( r ) + (cid:20) f (cid:48) ( r ) f ( r ) + 6 r (cid:21) ψ (cid:48) ( r ) + 1 r f ( r ) (cid:20) q φ ( r ) r f ( r ) − m (cid:21) ψ ( r ) = 0 . (10)Near the AdS boundary ( r → ∞ ), the spacetime geometry becomes the planar GB-RN-AdS blackhole with f ( r ) given as f ( r ) = 12 α (cid:34) − (cid:115) − α (cid:18) − r r (cid:19) + 3 αµ r (cid:16) r + r (cid:17) (cid:18) − r r (cid:19)(cid:35) . (11)Whereas, the asymptotic behavior of the matter fields are given by φ ( r ) = µ − ¯ dr ,ψ ( r ) = J C r ∆ − + Cr ∆ + , (12)where µ , ¯ d , J C , and C are regarded as the chemical potential, charge density, source, and the con-densate value (VEV) of the diquark operator dual to ψ , respectively, and the conformal dimensions∆ ± read ∆ ± = 12 (cid:18) ± (cid:113)
25 + 4 m l (cid:19) , l = 2 α − √ − α , (13)which suggests the Breitenlohner-Freedman (BF) bound[42, 43] as m l ≥ − . (14)Because the scalar field ψ is dual to the quark pair, the conformal dimension ∆ + of C should be∆ + = 2 × d − which is equal to four for the case of d = 6. This suggests m l = − − = 1. Note that, we need to impose the regularity condition for the matter fields at the eventhorizon as φ ( r + ) = 0 , ψ (cid:48) ( r + ) = r f (cid:48) ( r + ) ψ ( r + ) m . (15)Near the event horizon, the solution must have the following expansions f ( r ) = f + f ( r − r + ) + f ( r − r + ) + f ( r − r + ) + · · · ,φ ( r ) = φ + φ ( r − r + ) + φ ( r − r + ) + φ ( r − r + ) + · · · ,ψ ( r ) = ψ + ψ ( r − r + ) + ψ ( r − r + ) + ψ ( r − r + ) + · · · , (16)where f i , φ i , and ψ i (with i = 0 , , , · · · ) are constants. Because the function f ( r ) vanishes atthe event horizon, we find f = 0. Furthermore, the regularity condition (15) suggests φ = 0 and ψ = r f ψ m .The spacetime geometry dual to the confinement phase is the GB-AdS soliton solution [44]which is obtained via analytically continuing the planar GB-AdS black hole solution as ds = r (cid:0) − dt + h ij dx i dx j + f ( r ) dy (cid:1) + dr r f ( r ) , (17)where f ( r ) = 12 α (cid:34) − (cid:115) − α (cid:18) − r r (cid:19)(cid:35) , r = 25 R y , (18)with r = r to be a conical singularity of the GB-AdS soliton solution which is removed byimposing a period condition for the coordinate y . In this configuration of the spacetime geometry,the equations of motion for φ ( r ) and ψ ( r ) are obtained as φ (cid:48)(cid:48) ( r ) + (cid:18) f (cid:48) ( r ) f ( r ) + 4 r (cid:19) φ (cid:48) ( r ) − q ψ ( r ) r f ( r ) φ ( r ) = 0 , (19) ψ (cid:48)(cid:48) ( r ) + (cid:20) f (cid:48) ( r ) f ( r ) + 6 r (cid:21) ψ (cid:48) ( r ) + 1 r f ( r ) (cid:20) q φ ( r ) r − m (cid:21) ψ ( r ) = 0 . (20)The asymptotic behavior of the matter fields near the AdS boundary is the same as Eq. (12).Whereas, the solution near the tip r = r has the following expansions φ ( r ) = φ + φ log( r − r ) + φ ( r − r ) + φ ( r − r ) + · · · ,ψ ( r ) = ψ + ψ log( r − r ) + ψ ( r − r ) + ψ ( r − r ) + · · · . (21)We impose the Neumann-like boundary condition on the matter fields, i.e. φ = 0 and ψ = 0, toensure that their value is finite at the tip r = r . The boundary condition at the tip r = r for thematter fields are φ (cid:48) ( r ) = 2 q ψ ( r ) r f (cid:48) ( r ) φ ( r ) ,ψ (cid:48) ( r ) = − r f (cid:48) ( r ) (cid:20) q φ ( r ) r − m (cid:21) ψ ( r ) . (22)By using the expression of f ( r ) and the expansions of φ ( r ) and ψ ( r ), given in Eqs. (18) and (22),respectively, the above boundary condition becomes φ = 2 q ψ r φ ,ψ = − r (cid:18) q φ r − m (cid:19) ψ , (23)which suggests that Eqs. (19) and (20) allow the solution with φ ( r ) (cid:54) = 0. III. HOLOGRAPHIC CSC
As we discussed above, in the limit that the chemical potential approaches the critical value µ c , the backreaction of the scalar field is negligible. Since the bulk background configuration isdetermined by the following action S (cid:48) bulk = (cid:90) d x √− g (cid:20) R −
2Λ + (cid:101) α L GB − F µν F µν (cid:21) . (24)The spacetime metric solution dual to the confinement phase is given by the GB-AdS solitonmentioned in the previous section with the constant potential of the gauge field as φ ( r ) = µ. (25)Whereas, the spacetime metric solution dual to the deconfinement phase is given by the planar GB-RN-AdS black hole with the line element described by Eqs. (7) and (11), and the correspondingpotential of the gauge field is φ ( r ) = µ (cid:18) − r r (cid:19) . (26)The Hawking temperature of the planar GB-RN-AdS black hole is given by T = 14 π (cid:18) r + − µ r + (cid:19) . (27)The non-negative condition of the temperature suggests the suitable region for µ/r + as0 ≤ µr + ≤ √ . (28)Let us study the phase structure of the bulk background configuration by examining the freeenergy (in the canonical ensemble) of the planar GB-RN-AdS black hole and GB-AdS soliton.Using the result in Ref. [45], we can find the total on-shell Euclidean action for the EGB gravitycoupled to the U (1) gauge field in the present work as S E = (cid:20)(cid:0) r f (cid:1) (cid:48) (cid:0) r − αr f (cid:1) (cid:12)(cid:12)(cid:12) ∞ r + − l (cid:18) − αl (cid:19) r f (cid:0) r f (cid:1) (cid:48) (cid:12)(cid:12)(cid:12) ∞ − r φφ (cid:48) (cid:12)(cid:12)(cid:12) ∞ r + (cid:21) π r V T , (29)where V = (cid:82) dx dx dx . Then, we obtain the free energy of the planar GB-RN-AdS black holeand GB-AdS soliton as Ω BH = − r (cid:18) µ r (cid:19) π r V , (30)Ω Sol. = − r π r V . (31)Here, we see that the free energies of the planar GB-RN-AdS black hole and GB-AdS soliton inEinstein gravity and the EGB gravity are the same in the planar case although the solutions inthese two kinds of gravity are different. By comparing their free energy one find which configuration Confinement phase Deconfinement phase W BH > W Sol. W BH < W Sol. m T FIG. 1: The phase diagram for the confinement and deconfinement phases. is thermodynamically favored. The corresponding phase diagram is showed in Fig. 1. The criticalcurve (red one) which separates the configuration of the GB-AdS soliton and that of the planarGB-RN-AdS black hole is determined by the equation Ω BH = Ω Sol. .In the following, we study how the phase structure of the bulk background configuration, men-tioned above, changes when the scalar field condensate appears.
A. Deconfinement phase
Let us first study the necessary condition which destabilizes the scalar field and makes thecondensation occurring. From the equation of motion for the scalar field, one can find the effectivesquared mass m of the scalar field as m = m − q φ ( r ) r f ( r ) , (32)with f ( r ) and φ ( r ) given by Eqs. (11) and (26), respectively. The necessary condition which m breaks the BF bound is given as m < − l , (33)which leads to q φ ( r ) r f ( r ) > l . (34)0 F a= F a=- F a=- F a=- FIG. 2: The behavior of F ( z, ˆ µ, α ) as a function of z with various values of ˆ µ and α . The red, blue, green,purple, and orange curves correspond to ˆ µ = √ , 0 . × √ , 0 . × √ , 0 . × √ , 0 . × √ , respectively. The left-hand side of (34) can be rewritten as q φ ( r ) r f ( r ) = q αz (1 − z ) ˆ µ − (cid:113) − α (1 − z ) + α ˆ µ z (1 − z ) ≡ q F ( z, ˆ µ, α ) , (35)where z ≡ r + /r and ˆ µ ≡ µ/r + . The behavior of the function F ( z, ˆ µ, α ) in terms of z , ˆ µ , and α is shown in Figs. 2 and 3. We can see that F ( z, ˆ µ, α ) increases with either the growth of ˆ µ orthe decreasing of α at an arbitrary value of z . In particular, one find that F ( z, ˆ µ, α ) would getthe maximal value at ˆ µ = √ . In the region of α (cid:46) − .
56, the maximal value of F ( z, √ / , α )is about two and thus there is an upper bound as F ( z, ˆ µ, α ) <
2. In the remaining region of α , F ( z, √ / , α ) gets the maximal value at a point z max which depends on α and is found by solvingthe following equation ∂∂z F ( z, √ / , α ) = 0 , (36)1 F m (cid:239) = (cid:144) F m (cid:239) = · (cid:144) F m (cid:239) = · (cid:144) F m (cid:239) = · (cid:144) FIG. 3: The behavior of F ( z, ˆ µ, α ) as a function of z with various values of ˆ µ and α . The red, blue, green,purple, and orange curves correspond to α = 0 . − . − . − −
5, respectively. which leads to3 − α + 4 z (cid:2) (12 − z − z + 10 z ) α − (cid:3) + (4 z − (cid:112) − − z + 5 z ) α = 0 . (37)The corresponding maximal value F ( z max ( α ) , √ / , α ) as a function of α is numerically given inthe left panel of Fig. 4. As a result, we obtain the following equality0 < q φ ( r ) r f ( r ) < N c F ( z max ( α ) , √ / , α ) , (38)where we have used the relation q = N c . This result along with (34) leads to N c < (cid:115) α F ( z max ( α ) , √ / , α )1 − √ − α ≡ N ub c ( α ) . (39)More explicitly, we show the behavior of the upper bound for N c as a function of the GB couplingparameter α in the right panel of Fig. 4. We observe that N ub c decreases with the growth of α .In the case of α = 0 corresponding to Einstein gravity, we obtain the upper bound for the color2 - - - - a F H z m a x H a L , (cid:144) , a L - - - - a N c ub FIG. 4: Left panel: The maximal value F ( z max ( α ) , √ / , α ) as a function of α . Right panel: The upperbound for N c as a function of α . The horizontal dashed black lines correspond to the case of Einstein gravity. number N c as N c < √ (cid:39) .
89 [32]. In addition, N ub c in the EGB gravity with α > α > N c ≥
2. However, in the EGB gravity with α < N ub c is enhanced compared to Einstein gravity and increases as the amplitude of the GB couplingparameter α grows. As a result, the presence of the GB term can lead the scalar field condensatewith N c ≥ µ c , the CSC phase occurs due to the condensation ofthe scalar field which corresponds to the nontrivial solution of ψ as J C = 0 [to guarantee thespontaneous breaking of the U (1) symmetry in the system] and C (cid:54) = 0. We can obtain the criticalchemical potential µ c and thus the critical curve in the µ − T plane by solving numerically Eqs.(8)-(10) using the shooting method. In this method, the boundary values of φ and ψ can be derivedby setting their appropriate value near the event horizon. Of course, the critical chemical potential µ c and the critical curve depend on both the GB coupling parameter α and the color number N c .As analyzed above, the EGB gravity with α < N c ≥
2. Thus, we solve numerically Eqs. (8)-(10) with the negative GB coupling parameter to findthe critical chemical potential for N c ≥ µ c /r + and the slope of the critical line T c = T c ( µ c ) for various values of the GB coupling parameter α for N c = 2 and N c = 3 in Tables Iand II, respectively. It is found that as the amplitude of the GB coupling parameter α increases,the critical chemical potential µ c decreases for the event horizon r + kept fixed. This means thatthe condensation of the scalar field is easier to form with increasing the amplitude of α . This canbe understood as follows: the spacetime curvature or the gravitational attraction becomes weaker3 α µ c /r + T c /µ c − . . . − . . . − . . . − . . . − . . . − . . . µ c /r + and T c /µ c with various values of α at N c = 2. α µ c /r + T c /µ c − . . . − . . . − . . . − . . . − . . . − . .
985 0 . µ c /r + and T c /µ c with various values of α at N c = 3. if the GB coupling parameter α decreases, as seen in the behavior of the effective asymptotic AdSradius l eff which is a decreasing function of α ; in this sense as increasing the amplitude of α (with α <
0) the electromagnetic repulsion can be easier to overcome the gravitational attraction, whichresults in the formation of the scalar hair. In addition, one see that the larger amplitude of α leadsto the larger slope of the critical line T c = T c ( µ c ). This suggests that the region of the CSC phaseis larger, as seen in Fig. 5 for N c = 2 and N c = 3. On the other hand, increasing the amplitude of α makes the CSC phase more stable.In Fig. 5, we show the phase diagram when takes account of the backreaction of the scalarfield for N c = 2 (top panels) and N c = 3 (bottom panels) with various values of the GB couplingparameter α . The phase diagram is dramatically different from that in the case of the absenceof the scalar field, given in Fig. 1, with the presence of the critical line (the blue lines) in thedeconfinement region below which it represents the CSC phase which is dual to the the planarGB-RN-AdS black hole with scalar hair. (As we see later, the CSC phase does not exist in theconfinement phase with the values of α considered in Fig. 5.) The free energy of this configuration4 Confinement phase Deconfinement phase C S C m T Confinement phase Deconfinement phaseCSC m T Confinement phase Deconfinement phase C S C m T Confinement phase Deconfinement phase C S C m T FIG. 5: The phase diagram in the case of that the scalar field is taken into account for various values of N c and α . Top-left panel: N c = 2 and α = − .
5. Top-right panel: N c = 2 and α = − .
0. Bottom-left panel: N c = 3 and α = −
12. Bottom-right panel: N c = 3 and α = −
13. The regions below the blue lines refer tothe CSC phase. is given by Ω shBH = (cid:20) − r (cid:18) µ r (cid:19) − (cid:90) ∞ r + q r φ ψ f ( r ) dr (cid:21) π r V . (40)The second term in this expression is due to the condensation of the scalar field which is alwaysnegative and thus the free energy of the CSC state is always lower than that of the normal de-confinement state. In this way, above the critical chemical potential the CSC state contributesdominantly to the thermodynamics and since is thermodynamically favored. We conclude that the5 - - - - - a q m r FIG. 6: The depenedence of qµ/r in terms of the GB coupling parameter α . CSC phase in 4D YM theories (with N c ≥
2) can exists in the gravitational dual model with theframework of the EGB gravity with α <
0, which can not be found with the framework of Einsteingravity.
B. Confinement phase
In order to find the condensation of the scalar field in the confinement phase, we need to solveEqs. (19) and (20) in the background of the GB-AdS soliton. First, let us determine the necessarycondition which corresponds to the breakdown of the BF bound as q φ ( r ) r > l , (41)with φ ( r ) given in Eq. (25), which leads to qµr > (cid:115) − √ − α α . (42)In the case of Einstein gravity, we derive qµ > . r = 1. The behavior of qµ/r as a functionof the GB coupling parameter α is shown in Fig. 6.The sufficient condition for the condensate of the scalar field in the confinement phase canbe obtained by solving numerically Eqs. (19) and (20) in the GB-AdS soliton background. Thecorresponding numerical values of the rescaled critical chemical potential qµ c are given in TableIII. With α = − . α = − .
0, the values of the critical chemical potential µ c are 1 . . N c = 2. Because the confinement phase exits at the chemical potentialwhich is below 1 .
73, the CSC phase does not exist in the confinement phase with these values of6 α qµ c α qµ c . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . − . . qµ c with various values of α in the confinement phase. α , as seen in the top panels of Fig. 5. This happens similarly to the case of N c = 3 with α = − α = −
13. In addition, from Table III we find that the rescaled critical chemical potential qµ c decreases with the increase of the amplitude of the GB coupling parameter α . This implies thatfor the sufficiently large amplitude of α , the critical chemical potential µ c would be lower than1 .
73 and since the CSC phase can appear even in the confinement phase. This may indicate thebreakdown region of the GB term in investigating the CSC phase transition.
IV. CONCLUSION
The quark matter at sufficiently high chemical potential and low temperature is expected toexhibit a color superconductivity (CSC) phase which might be present in the cores of neutronstars. In Ref. [32], a bottom-up holographic model was introduced in the framework of Einsteingravity to describe the CSC phase in the Yang-Mills (YM) theory. Based on the analysis wherethe backreaction of the matter part is considered and thus improves the results of the probeapproximation, the authors have concluded that the CSC phase appears in the deconfinementphase but not the confinement one for the color number N c = 1. However, for N c ≥ N c ≥
2. We consider a gravitational system with the matter content consisting of a U (1) gauge fieldand a charged scalar field in the framework of Einstein-Gauss-Bonnet (EGB) gravity. Here, thegauge field and scalar field are dual to the current of the baryon number and the diquark operator7in the boundary field theory, respectively. We have indicated that the Gauss-Bonnet (GB) termplays a role in the breakdown of the Breitenlohner-Freedman (BF) bound and thus the scalar fieldcondensate, corresponding to the occurrence of the CSC phase.Near the critical chemical potential, the scalar field condensate approaches zero and hence itsbackreaction on the spacetime geometry is negligible. As a result, the bulk background configura-tion is given by the EGB gravity coupled to the U (1) gauge field in the asymptotic AdS spacetime.The deconfinement and confinement phases are dual to the planar GB-RN-AdS black hole andGB-AdS soliton, respectively. We have calculated their free energy in the canonical ensemble andobtain the corresponding phase diagram.When taking the scalar field into account, we study the scalar field condensate and its modifi-cation on the phase structure of the bulk background configuration in both the deconfinement andthe confinement phases. We determine the necessary condition for destabilizing the scalar field andmaking it condensing by examining the breakdown of the BF bound. In the deconfinement phase,we found that the CSC phase for N c ≥ α >
0. However, with α <
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