aa r X i v : . [ h e p - t h ] O c t Sakai-Sugimoto Brane System at High Density
D. Yamada
Racah Institute of Physics , The Hebrew University of Jerusalem , Givat Ram, Jerusalem, 91904 Israel [email protected]
Abstract
The D4-D8 brane system of Sakai-Sugimoto model at high quark density is studied inthe weak coupling regime. We show that the color superconducting phase (for N c ≈ N c → ∞ ) disappears at very large chemical potential,or equivalently at very large compactified dimension that the model possesses. We alsocomment on the prospects in the strong coupling regime along with the QCD phasediagram. Contents Introduction
Decades have passed since quantum chromodynamics was accepted as the theory of stronginteraction. While the successes of this theory are impressive, there still remain numerousunresolved problems. The most prominent of those problems is the confinement. At lowenergy, we observe hadrons instead of the QCD fundamental degrees of freedom and we stilldo not quite understand the mechanism of this phenomenon. The lack of the understandingis mainly due to the strongly coupled nature of the theory at low energy and to the fact thatwe do not have very good analytical control over the field theory in such coupling regime.The best hope, therefore, is that the numerical study could provide important insights intoQCD at strong coupling.Confinement is one of the phases of QCD and there are other phases in different re-gions of the QCD parameter space. One such example is the deconfined phase at hightemperature. The numerical method, in fact, is proven to be powerful in the investigationof the confinement/deconfinement phase transition, and has provided the estimates of thetransition temperature and the order of the phase transition. (See Reference [1] and thepapers cited therein.) There is, however, a large region of the parameter space in whicheven the numerical study has not been very successful. Namely, the theory at finite quark(baryon) density. There is a problem in carrying out lattice simulations with the nonzerochemical potentials that are conjugate to the density and this is known as the sign problem(see e.g.
Reference [2]). While the effort to overcome the sign problem vigorously continues,there have been some developments in analytic approaches. These include the analysis atasymptotically large chemical potential and also the use of NJL type models. The formertakes the advantage that the QCD coupling is expected to be weak at very high density andperturbative computations from the QCD Lagrangian itself are possible. The latter modelsare pure fermionic and quartic couplings, which reproduce properties of QCD interactionsin a certain degree, are introduced. What has been emerging from these analyses is the very rich phase structure of QCDin the parameter space of the temperature and the chemical potential. (See, for example,Reference [8] for the phase diagram.) In general, a cold and highly dense quark matteris expected to become a color superconductor [9]. As mentioned before, when the quarkchemical potential µ is very large, the coupling g ( µ ) is small and the excitations near theFermi surface of the quarks, particles and holes, are nearly free, and this naive groundstate at high density is known as the Fermi liquid. However, the pairs of the particles atthe antipodal points of the Fermi sphere are all degenerate and it costs no free energy toform such a pair. Then, if there is an attractive force between the particles (or holes),the pairing actually reduces the free energy of the system, leading to the instability of thenaive ground state against the formation of the pair (Cooper pair). This is known as theBardeen-Cooper-Schrieffer (BCS) instability [10] and in their original work for the electrongas in a solid, the attractive force was provided by the phonon exchange. For the case ofhigh density QCD, there is an attractive force in a color channel and it leads to the similarBCS instability.In particular, if we have three massless flavors in the theory, a very interesting form of A good modern review on perturbative high density QCD is Reference [1]. The NJL-model was intro-duced by Nambu and Jona-Lasinio [3, 4] as the model that exhibits the chiral symmetry breaking and lighthadronic spectrum. A good review on the NJL model is Reference [5]. The model is applied to QCD atfinite density and reviewed in References [6, 7]. This is the (scalar) diquark condensate of theform h q aLi q bLj i = −h q aRi q bRj i = ∆ CF L ( δ ai δ bj − δ aj δ bi ) , (1)where the superscripts a, b are the color indices, the subscripts i, j are for the flavors, thesubscripts L, R indicates the chirality of the quarks and ∆
CF L is the size of the condensate(the gap). As one can see, the Kronecker deltas relate the flavor and color symmetriesand those are not separately preserved by the condensate. The residual symmetry is thesimultaneous flavor and (global) color rotations and for this reason, this phenomenon iscalled the color-flavor locking (CFL). The full symmetry breaking pattern of CFL is SU (3) color × SU (3) L × SU (3) R × U (1) V × U (1) EM → SU (3) color+ L + R × Z × U (1) ˜ Q , (2)where SU (3) color+ L + R is the global diagonal subgroup of the original color and flavor sym-metries, Z is the subgroup of U (1) V that changes the sign of all the quarks and U (1) ˜ Q isknown as the “modified electromagnetism” whose gauge boson is a linear combination ofthe original photon and one of the gluons [11, 1]. We observe that since L - and R -flavorsymmetries both lock to the color, the chiral symmetry is broken through the color factor.Even though the mechanism of the chiral symmetry breaking is very unusual, the corre-sponding chiral Lagrangian can be built [12]. This novel phase of QCD created a renewedand wide interest in the QCD phase structure and many generalizations and modificationshave been explored. For example, when one considers finite quark masses, there are otherpossible forms of the condensate and those can be energetically favored for some regions ofthe µ -T parameter space, resulting in the complicated structure of the phase diagram. Theinterested reader can pursuit the subject in the review papers cited above.Rather than continuing to overview the QCD phase structure, we would now like toturn to the high density behavior in the ’t Hooft limit with weak coupling. (This is notQCD, which has N c = 3, but has a potential relation to the holographic theories.) Inthis case, we do not expect the color superconductor to be the correct ground state ofthe cold QCD. Heuristically, this is because the Cooper pair is not a color singlet and notexpected to survive the limit. The possibility of a color singlet condensate of particle andhole (not anti-particle) has been investigated by Deryagin, Grigoriev and Rubakov (DGR)in Reference [13]. When the particle and hole at the antipodal points of the Fermi sphereform a pair, the condensate is not homogeneous nor isotropic but is a standing wave in acertain direction; h ¯ q L ( x ) q R ( y ) i = e i~p F · ( ~x + ~y ) f ( x − y ) , (3)where | ~p F | = µ and f is a function that describes the amplitude of the standing wave. Thecondensate is called the chiral density wave ( χ DW) and it breaks the chiral symmetry butnot the gauge symmetry (in the limit x → y ). This condensate is kinematically less favoredthan the Cooper pairing at N c ≈
3, but it has been shown that the ground state instabilitydue to the formation of χ DW (DGR instability) dominates over the BCS-type instabilityin the large N c limit [13, 14].The aim of this paper is to examine the high density behavior of the Sakai-Sugimotomodel [15, 16]. As we will describe in Section 2, this is a model in Type IIA string theorywith a certain brane configuration. What makes this model interesting is that the low This model with N f = 3 massless quarks is an approximate QCD where the masses of u, d, s are set tozero and those of c, b, t are taken to infinity. N c ≈ N c → ∞ , respectively, are absent and the ground state is the Fermi liquid. InSection 3, we comment on the known finite density analysis of the model at strong coupling,contrasting to the QCD expectations. In the strong coupling analysis, the possibilities ofthe superconductivity and χ DW have not been addressed and we discuss some prospects inthis direction.
In this section, we discuss the Sakai-Sugimoto brane system at high density with the weakYang-Mills coupling of the world-volume theory. Though the results are relatively straight-forward, the computations are somewhat involved. We therefore split the discussion intothe qualitative and quantitative parts. In the first part, we qualitatively explain the highdensity behavior of the system, then in the second part, we carry out the computations andconfirm the qualitative expectations.
Let us briefly review the BCS and DGR instabilities of high density QCD in a way thatwould provide the conceptual background for the quantitative calculations. For simplicity,we set all current quark masses to zero, which is a good approximation when the quarkchemical potential is much larger than the mass of the heaviest quark. In the presence ofthe quark chemical potential, µ , at zero temperature, we have a well-defined Fermi sphereof radius µ . As a convention, we take the excitations near the Fermi surface be particles asopposed to anti-particles. When µ is very large, the anti-particles are buried deep in theDirac sea and will not play a role in the following discussion.On the Fermi surface, the free energy of the states are zero (more precisely, at the mini-mum) and it costs no free energy to change momentum along the Fermi surface. Therefore,the energy scales only in the radial direction of the sphere and we consider the renormaliza-tion group flow as we scale the energy down toward the Fermi surface. The relevant degreesof freedom are the particles and holes near the Fermi surface and we are interested in the For a modern exposition of the BCS instability, see Polchinski’s TASI lecture notes [17]. ~p F with the magnitude | ~p F | = µ and decompose afour-momentum near this point as p ν = ( E, ~p F + ~l k + ~l ⊥ ) , (4)where ~l k is parallel to ~p F and ~l ⊥ is perpendicular to it. As stated before, only l k scales withenergy and l ⊥ may be trivially integrated along the Fermi surface in a diagram computation.We, therefore, have the 1 + 1-dimensional effective theory that describes the dynamics ofthe particles and holes. It is important to notice that the kinematics is restricted becausethe dynamics must take place near the Fermi surface. When we consider a particle-particleor hole-hole scattering, it is clear that the scattering must be near back-to-back, that is,the scattering pairs must be at antipodal points of the Fermi sphere. In the back-to-backscattering, the scattering angle may be arbitrary without spoiling the kinematic restrictionand hence the phase space of this scattering is all over the Fermi surface.Now in a two-dimensional theory, irrelevant operators of four dimensions may becomerelevant or marginal. In particular, a four-fermion interaction is marginal in two dimen-sions. If the interaction is attractive, the quartic coupling grows as we scale the energydown toward the Fermi surface and it eventually hits the Landau pole. This implies thatthe perturbation theory breaks down at the infrared scale around the pole. For the scat-tering of particle or hole pairs, this indicates that the naive ground state of the weaklyinteracting particles and holes near the Fermi surface, the Fermi liquid, is unstable againstthe formation of Cooper pairs. This is the BCS instability, and the new ground state has agap due to the formation of the condensate whose size is roughly the location of the Landaupole. Technically, as we will see in the next subsection, the instability is closely relatedto the infrared divergence that appear in the perturbation theory and the gap properlyprovides the infrared cutoff. In weakly coupled QCD, i.e. , QCD at very high density, theleading order contribution to the interaction is given by the one-gluon exchange and it isattractive in the antisymmetric ¯ -channel, resulting in the color superconductivity.We now turn to the DGR instability. This is associated with the scattering of the particleand hole located at the antipodal points of the Fermi sphere. In this case, the scatteringis not back-to-back, but near forward. Unlike the back-to-back case, the scattering anglecannot be too large to stay near the Fermi surface and the phase space of the forwardscattering is limited to a very tiny patch on the Fermi surface. The difference in the phasespaces for the BCS and DGR cases will be important in the quantitative computations.Now the particle-hole forward scattering is similar to the Bhabha scattering whose am-plitude has the forward enhancement. Thus also in our case, we can expect an infrareddivergence (the DGR instability) as the exchange gluon becomes very soft, leading to acondensate of the particle-hole pair ( χ DW). This, however, does not happen in weakly cou-pled QCD. The reason is that the finite density screening effect completely overwhelms sucha condensate; it is the screening that provides the infrared cutoff and not the formation ofa condensate.The situation is different in the large N c limit (with small ’t Hooft coupling). First,note that the Cooper pair of the BCS instability is not color singlet while the χ DW ofDGR is. Therefore, the BCS instability is 1 /N c suppressed and DGR is not. Secondly, A rigorous derivation of the high density effective theory of QCD was carried out by Hong [18]. U (1) V -symmetry, the finite density screeningeffect is provided by the quark loops, such as the one shown in Figure 1, and the gluonloops do not contribute because the gluon propagator is independent of the quark chemicalpotential. This implies that the finite density screening is suppressed as the number of theFigure 1: One loop diagram that contributes to the finite density screening. color is taken to a large value. In fact, the DGR instability was discovered by disregardingthe the screening effect and the authors of Reference [13] noted that the χ DW can formand dominate over the Cooper pairing at least in the large N c limit. Later, Shuster andSon showed that the DGR instability may occur if N c & N f , where N f is the numberof the flavor [14].This is an example where the large N c limit of QCD yields qualitatively different prop-erties. In weakly coupled QCD, χ DW is not a relevant phenomenon. Though it couldpossibly compete with the Cooper pair at strong coupling, so far the situation is unclear[19].
We now consider whether the Sakai-Sugimoto model at high density exhibits similar prop-erties as discussed above. For this purpose, we assume that the model is in the regime withlow energy and weak Yang-Mills coupling so that we can use the perturbative world-volumefield theory arguments.The model is Type IIA string theory with D4-, D8- and D8-branes. The configuration ofthe branes is shown in Figure 2. The x -direction is compactified to the circle of circumfer-ence L and the D8D8-branes are placed at the antipodal points of the circle. To discuss thelow energy spectrum of the model, we first consider only the compactified N c D4-branes. InReference [20], Witten suggested that if we impose the anti-periodic boundary condition tothe adjoint world-volume fermions, the low energy world-volume theory has the spectrumof 3 + 1-dimensional pure Yang-Mills theory. This is because the fermions get tree levelmass of order 1 /L and the scalars, including the compactified component of gauge field, A ,get the one-loop mass of order g/L , where g := g / √ L and g := (2 π ) g s l s , with g s stringcoupling and l s string length scale, and we assume g ≪ Now, Sakai and Sugimoto insert the N f D8 and N f D8 branes as shown. As explainedin Reference [21], there are massless fermions at the 3 + 1-dimensional intersection of D4-and D8-branes. These are the lowest states of the strings stretching from D8 to D4. Sincethe world-volume U ( N f ) L gauge symmetry of D8-branes acts as the flavor symmetry, thesemassless fermions are fundamental “quarks”. Similar massless fermions are also present Unlike four-dimensional case, other components of the gauge field do not acquire mass of order g /L ,in all orders of the perturbation theory.
40L L/2 x Figure 2:
Sakai-Sugimoto brane configuration. The x -direction is compactified with period L . The D4world-volume fermions have the anti-periodic boundary condition. We locate the N f D8-branes and N f D8-branes at x = 0 = L and x = L/
2, respectively. at the intersection of D4- and D8-branes. However, the GSO projection projects out op-posite chiralities to those fermions at different intersections. Therefore, we call the mass-less fermions at D4D8 and D4D8 intersections as “left-handed ( q L ) and right-handed ( q R )quarks”, respectively.This theory at low energy, therefore, has U ( N f ) L × U ( N f ) R flavor symmetry and thetheory appears to be very similar to the massless QCD, if N c and N f are appropriatelychosen. The difference, of course, is that the fermions with different chiralities are separatelylocated in the x -direction and the gluons propagate in five dimensions, including the x -direction. As argued by Antonyan et al. [22], this QCD-like dynamics of the quarks at theintersections does not change even if the period L is large; the shift symmetry of the D4adjoint scalars and the five dimensional gauge symmetry allow the scalars, including A , tocouple to the fundamental fermions only through derivative interactins which is suppressedby the string scale.In the weak coupling analysis that we carry out in this section, we assume that theperiod L is much larger than the string length scale l s so that the tachyon becomes heavyand decouples. Also as we have already explained, we place the D8 and D8 branes at theantipodal points of the compactified circle.In examining this model, we need to decide on how the quarks interact through theexchange of the gluons that propagate in the x -direction. One possibility is the non-localinteraction. This scenario takes only into account of the zero-mode of the discrete mo-mentum in the x -direction. In this case, the theory becomes completely insensitive tothe existence of the fifth dimension and behaves in the same way as the four dimensional When the D4-branes of the system are replaced with their effective geometry (which is not the case in ourdiscussion), the proper distance of the D8s becomes less than the string scale in the region sufficiently nearthe horizon and the statement made here is no longer valid in the analysis with the background geometry.Refs.[23, 24, 25] show that the tachyon indeed condenses and it is responsible for the chiral symmetrybreaking of the model at strong coupling. L is large. We thus take the second alternative where the D8-branes aretreated as sources (or stiff walls) and allow the exchanged gluons to carry arbitrarily highmomenta in the x -direction. Such an assumption is reasonable because the D8 branes areinfinitely heavier than the D4s and consistent with the fact that we are treating the D8sas the flavor branes, that is, we are neglecting their fluctuations. The momentum is notconserved in the x -direction but this is natural in that the translation symmetry is brokenin this direction.We now explain how we introduce the quark chemical potential. We have the globalsymmetry SU ( N f ) L × SU ( N f ) R × U (1) V . The chemical potential that we are interested in isconjugate to the U (1) V charge. The standard way to introduce a chemical potential in a fieldtheory is to treat it as the constant background “gauged” field of a U (1) global symmetry,with all the components being zero except the time component. In this way, the chemicalpotential modifies the time component of the covariant derivatives in the Lagrangian of thetheory. Thus in our case, the simplest way to introduce the quark chemical potential is toturn on the A constant background gauge fields of U (1) ∈ U ( N f ) L and U (1) ∈ U ( N f ) R world-volume gauge symmetries and tune them to an equal value. Actually, the backgroundfields may not be constant all over the D8-branes and the only requirement is to have theconstant value at the intersections with the D4-branes. Thus, for example, we may turn onthe field that depends on the radial direction in the 5,6,7,8 and 9 directions which wouldcorrespond to the nonzero electric field in the world-volume.What we will find in the quantitative analysis of the next section is rather intuitive.As we have mentioned, if we consider the effect only of the zero-mode momentum in the x -direction, the theory reduces to the QCD-like theory. Thus when the compactificationscale 1 /L is very large compared to the energy scale of the interest, in this case it is the valueof the chemical potential µ , we expect to have the BCS and DGR instabilities at N c ≈ N c → ∞ , respectively, just as described before. Now as the compactification scale1 /L gets smaller, the infrared effect in the x -direction becomes comparable to the one thatleads to the regular BCS or DGR instability. As a consequence, the size of the condensategrows and eventually becomes too large to maintain the dynamics near the Fermi surface.Therefore, when the scale 1 /L is small with respect to µ , there is no BCS or DGR type ofinstability and the ground state of the theory is described by the Fermi liquid.In this qualitative discussion, it is not clear at what scale this crossover occurs. Thecomputations of the next subsection show that at N c ≈ µL & /g , no BCS-typeinstability is present and at N c → ∞ and µL & e / √ λ / √ λ with λ := g N c , no DGR-typeinstability happens. Notice that the DGR instability persists to exponentially larger valueof µL compared to the BCS case. This is because the phase space of the particle-holescattering is very small, in fact it is exponentially small, and the discrete momentum in thecompactified direction must become as fine as this scale to open up the extra dimension.The situation explained here is schematically summarized in Figure 3. The gluon momenta actually have to be cut off below the string scale to avoid the derivative interactionsbetween the D4 adjoint scalars and the fundamental fermions. However, as we will see shortly, the highmomenta increasingly suppress the gluon propagator and their contributions become negligible at sufficientlyhigh scale. Therefore we can approximately take the x -momentum to infinity. BCS DW χ Fermi LiquidNc L µ Figure 3:
A schematic phase diagram of the theory at weak coupling, high density and zero temperature.Being schematic, the straight lines may not be straight nor sharp transitions in reality.
We now demonstrate quantitatively what has been discussed in the previous subsection.We carry out the renormalization group and Dyson-Schwinger analyses. The former ismore intuitive in accordance with the qualitative discussion and shows the existence of theinstabilities. But this method does not provide the size of the gap and this is augmentedby solving the Dyson-Schwinger equations.We adopt the conventions of Wess and Bagger [26], except the definition of the Diracspinor; ψ := (cid:18) q Lα q ˙ αR (cid:19) , ¯ ψ := (¯ q αR , ¯ q L ˙ α ) . (5)We mainly work in the chiral basis. As a convention, the undotted and dotted spinors liveon the D8 and D8 branes, respectively.Our central focus of this subsection is to show the existence of the instabilities and toobtain the size of the gap. We are less interested in the exact color-flavor structure of thecondensate, so in what follows, we simplify the analysis by suppressing the flavor structure.This is similar to N f = 2 case where the Pauli principle requires the simpler quark pairing. Our first analysis is macroscopic in a sense that we introduce an effective one point four-fermion coupling. Then we observe how the effective coupling evolves as we scale the energyof the system down to the Fermi surface. This idea was first carried out in high densityQCD by Evans et al. in Reference [27].Since we are dealing with the weak coupling at high density, the quark interaction canbe approximated by a single gluon exchange. We then model the four-fermion interactionby replacing the one-gluon exchange to a point, as shown in Figure 4. Because the chemicalpotential breaks the 3 + 1 world-volume symmetry down to O (3), we separately handle thecouplings, G and G j , as in iG ( ¯ ψγ ψ ) , iG j ( ¯ ψγ j ψ ) . (6)9igure 4: Replacing the one-gluon exchange to an effective one point interaction.
For the one-gluon exchange, we can further write G ( D ) = − G j ( D ) := − g X ( D ) F . (7)Notice that we have included the minus sign from the signature in the definition of G . Theconstant g is the five dimensional Yang-Mills coupling as before, F is the form factor thatarises from the gluon propagator and X ( D ) := 12 { C ( D ) − C ( (cid:3) ) } , (8)with C ( D ) being the Casimir operator of SU ( N c ) in the representation D and (cid:3) being thedefining representation.We consider three color channels; symmetric (symm), antisymmetric (asymm) and sin-glet ( • ). For those cases, we have X (symm) = N c − N c , X (asymm) = − N c + 12 N c , X ( • ) = − N c − N c , (9)where we have adopted the normalization tr( T α T β ) = (1 / δ αβ for the N c × N c matrices { T α : α = 1 , . . . , N c − } of the defining representation. Notice the large N c behavior ofthose factors. The singlet channel is larger than the other channels by a factor of N c inthe absolute value. This can be easily understood in the double line notation of the singlegluon exchange diagrams, as shown in Figure 5.Figure 5: The double line notations of the one-gluon exchange. The top and bottom sets represent thesymmetric (or antisymmetric) and singlet channels, respectively. If we fix the colors of the incoming quarks,the symmetric channel has the fixed colors for the scattered quarks, while the singlet channel has N c choices. Let us now consider the form factor, F . The gluon propagator in the Feynman gauge hasthe form 1 / ( p + p ) with the discrete momentum, p = 2 πn/L , in the compactified directionand this propagator suffers from an infrared divergence. To cure this problem, we introduce10n infrared cutoff m . We simplify the situation by assuming that m is the same for the timeand spatial components of the gluons. Possible one-gluon interactions are illustrated inFigure 6. In both cases shown in the figure, the discrete momentum is 2 πn/L because this
D8 D8a b a b0,12,3 4 D8 D8D8
Figure 6:
One-gluon exchange diagram. The gluons are propagating in the compactified x -direction. TheD8D8-branes are treated as sources or stiff walls. is determined by the period of the compactification and not by the distance between thebranes. Therefore, the form factor, that arises from the gluon propagator, is independentof the D8 brane distance and is the same for the both interactions in Figure 6 (includingthe case with the gluons making many rounds in the compactified circle). We follow Evans et al. [27] to obtain the form factor, namely, we take average of the gluon propagator overthe scattering phase space. For the dynamics very near the Fermi surface, the energy andthe component of the momentum in the radial direction of the Fermi sphere are almostzero. Hence apart from the extra x -direction, the phase space is two dimensional along theFermi surface. Let p and p be the incoming and outgoing momenta of a scattering quark.We then have the gluon momentum p = ( p − p ) ≈ µ (1 − cos θ ) , (10)where θ is the scattering angle.For the back-to-back scattering, the angle ranges from 0 to π and the phase space is allover the Fermi surface. Therefore, the form factor in this case is F BB = 1 N L ∞ X n = −∞ Z d pp + (2 πn/L ) + m = 2 π N L ln (cid:20) sinh (cid:18) mL p µ /m (cid:19) / sinh (cid:18) mL (cid:19)(cid:21) , (11)where we have defined the total phase space factor N := 4 πµ . We have already carriedout the sum over the p -discrete momentum because the p dependence is only in the gluon At very high density where the coupling is weak, one can expect the finite density Debye screening oforder gµ provides the infrared cutoff. This is true for the time component but not for the spatial components.There is no static screening in the spatial (magnetic) components and the magnetic screening is dynamicaldue to the Landau damping. This was pointed out by Son [28] and he discovered that the dynamicalscreening effect leads to the qualitatively different form of the gap. We will take into account of Son’s effectin the Dyson-Schwinger analysis. θ UV , where the latter angle is much lessthan π and limits the phase space to a little patch on the Fermi surface. The form factorthen takes the form F F W = 1 M L ∞ X n = −∞ Z d pp + (2 πn/L ) + m = 2 π M L ln (cid:20) sinh (cid:18) mL p − cos θ UV ) µ /m (cid:19) / sinh (cid:18) mL (cid:19)(cid:21) , (12)where M := 2 πµ (1 − cos θ UV ). Notice that since θ UV ≪ π , we have M F F W < N F BB . Thisfact will lead to the dominance of the BCS-type instability over the DGR type for N c ≈ mL is small, the form factors behave logarithmicallywith respect to the parameter µ/m and they behave linearly when mL is large. This isbecause when the the compactified dimension is very small, the contributions from n = 0is also small and the integral is effectively two dimensional, leading to the log behavior.When the compactification size is very large, the discrete momentum becomes finer and theintegral essentially becomes three dimensional and the form factors behave linearly.Having modeled the four-fermion interaction, we now derive the renormalization groupequations for the couplings. The diagram that drives the renormalization group flow is thefermion one-loop diagram and we consider the three cases shown in Figure 7. For Diagram (a) (b) (c)LL L LR R Figure 7:
The diagrams that drive the renormalization group flow. The letters “ L ” and “ R ” refer to left-and right-handed quarks, respectively. (a) in the back-to-back scattering, we can deduce from the expressions (6) that each vertexcorresponds to either iG LL ¯ σ αα ¯ σ ββ , or iG jLL ¯ σ j ˙ αα ¯ σ j ˙ ββ . (13)When the both vertices correspond to G LL , the diagram yields( iG LL ) (¯ σ δα ¯ σ γβ ) Z d p (2 π ) " − i ( p ν − µδ ν, ) σ να ˙ α ( p λ − µδ λ, ) − i ( − p η − µδ η, ) σ ηβ ˙ β ( − p λ − µδ λ, ) (¯ σ αδ ¯ σ βγ ) , (14)where we used the quark propagators shown in Appendix A. Note that the momentumintegral is four dimensional rather than five. The dynamics of the quarks are restrictedto the four dimensional intersections of the branes and the information about the extradimension has been encoded in the form factor. We decompose the momentum as in Equa-tion (4) but redefine ~p F to include ~l ⊥ . Then near the surface of the large Fermi sphere, the12ectors ~p F and ~l k are near parallel and we also have E, l k ≪ µ and p F ≈ µ . Under theseapproximations together with the use of the O (3)-invariance, the argument of the squarebracket in Equation (14) becomes − (cid:18) − σ α ˙ α σ β ˙ β + 13 σ j α ˙ α σ j β ˙ β (cid:19) E − l k . (15)One can integrate over E , either by the contour integral or by the Wick rotation, then | l k | is integrated from the scale Λ UV down to Λ IR . One can also simplify the σ -matrices (all thenecessary formulas are given in Appendix B of Wess and Bagger [26]) and the expression(14) becomes i N π ( G LL ) (cid:18) ¯ σ δδ ¯ σ γγ −
13 ¯ σ j ˙ δδ ¯ σ j ˙ γγ (cid:19) t , (16)where we have defined t := ln(Λ IR / Λ UV ). This parameter t has the range ( −∞ ,
0) and thelower limit corresponds to the Fermi surface. When one of the vertex of Diagram (a) is G and the other is G j , similar procedure yields i N π ( G LL G jLL ) (cid:18) − σ δδ ¯ σ γγ + 103 ¯ σ j ˙ δδ ¯ σ j ˙ γγ (cid:19) t , (17)and when the vertices are both G j , we get i N π ( G jLL ) (cid:18) σ δδ ¯ σ γγ −
133 ¯ σ j ˙ δδ ¯ σ j ˙ γγ (cid:19) t . (18)From those results, we obtain the renormalization group equations dG LL dt = N π n − ( G LL ) + 2 G LL G jLL − G jLL ) o ,dG jLL dt = N π (cid:26)
13 ( G LL ) − G LL G jLL + 133 ( G jLL ) (cid:27) . (19)These equations can be diagonalized to the following forms d ( G LL − G jLL ) dt = − N π ( G LL − G jLL ) , (20) d ( G LL + G jLL ) dt = − N π ( G LL + G jLL ) . (21)One can carry out the same procedure for the back-to-back scattering of Diagram (b) inFigure 7 and obtain d ( G LR + 3 G jLR ) dt = 0 , (22) d ( G LR − G jLR ) dt = − N π ( G L ¯ R − G jL ¯ R ) . (23)Above four renormalization group equations are obtained by Evans et al. [27] (but withdifferent form factors). 13he Diagram (c) of Figure 7 is similar to the case with (a) [and not with (b) because ¯ q R has an undotted spinor index just as q L ]. But we should recall that we are interested in theforward scattering in this case. Therefore in the loop, the top left-handed quark propagatorcarries the momentum ( E, ~p F + ~l k ) and the bottom right-handed one carries ( E, − ~p F + ~l k )in the directions of the arrows in the quark loop. Then the propagator part of the diagramcorresponding to (15) of the Diagram (a) is − (cid:18) σ α ˙ α ¯ σ ββ − σ jα ˙ α ¯ σ j ˙ ββ (cid:19) E − l k . (24)This structure is the same as the LL -case (a), except the overall sign. This sign is cancelledby the other one that comes from the difference in the direction of the quark line. Thus,apart from the integration range of the scattering angle θ , the cases (a) and (c) are thesame. So we have for Diagram (c), d ( G L ¯ R − G jL ¯ R ) dt = − M π ( G L ¯ R − G jL ¯ R ) , (25) d ( G L ¯ R + G jL ¯ R ) dt = − M π ( G L ¯ R + G jL ¯ R ) . (26)Now for the generic form of the renormalization group equation dG ( t ) dt = − KG ( t ) , (27)with some constant K , the Landau pole, if exists, is reached at t LP = − KG (0) . (28)Since the range of t is ( −∞ , KG (0) is positive andthe larger the factor KG (0) is, the faster the pole is reached. We note that because ofthe constants N and M in the form factors (11) and (12), the Landau pole is independentof those. From Equations (7) and (9), and also with the fact that M F F W < N F BB ,we see that the instability is dominated by the BCS type in the LL , color antisymmetricchannel for N c ≈
3. Note that the color symmetric channel does not have instabilitybecause the interaction in this channel is repulsive. When N c is sufficiently large, the L ¯ R -channel, whose coupling is proportional to the product of X ( • ) and F F W , dominates overthe other channels because the inequality M F F W < N F BB is compensated by the fact that | X (asymm) | < | X ( • ) | . Thus the DGR-type instability dominates over BCS in this regime.In our method here, it is not possible to estimate the value of N c at which the crossoverfrom the BCS- to DGR-type instability occurs, because we crudely introduced the commoninfrared cutoff, m , to all the components of the gluon propagator and we do not have theactual value of θ UV . See Shuster and Son [14] for the estimate of the value N c .Recall that the form factors grow linearly when the parameters mL and µ/m are largeand the effective four-fermion couplings become large accordingly. We thus expect that thewhole analysis breaks down when those parameters are exceedingly large, and we need toresort to the microscopic analysis, i.e. , the analysis with the fundamental interactions, togain insight into the nature of the pathology. This is the subject of the next analysis.14 .2.2 Dyson-Schwinger Equations We now turn to the analysis based on the Dyson-Schwinger equations. The traditional formof the Dyson-Schwinger equation in the diagrammatic representation is shown in Figure 8.This method is less intuitive compared to the previous renormalization group analysis, but = Figure 8:
The diagrammatic representation of the Dyson-Schwinger equation for the diquark condensate.The triangle denotes the gap insertion. The square in the gluon propagator represents the screening effect. it lets us obtain the actual size of the gap. This analysis is microscopic which deals withthe quarks rather than the quasi-particles of the effective theory and interactions are theQCD interactions rather than the effective ones.As usual, we adopt the Nambu-Gor’kov formalism (see, e.g. , Reference [9]). In order tointroduce the Nambu-Gor’kov basis, we define the charge conjugate Dirac spinors as ψ C := C ¯ ψ T and ¯ ψ C := ψ T C T where the charge conjugation matrix C is defined in Appendix A.In the Weyl basis, these can be expressed as (cid:18) ¯ q cRα ¯ q c ˙ αL (cid:19) = ( iσ α ˙ β ¯ σ βγ )¯ q Rγ ( i ¯ σ αβ σ β ˙ γ )¯ q ˙ γL ! , ( q c αL , q cR ˙ α ) = (cid:16) q Lγ ( − iσ γ ˙ β ¯ σ βα ) , q R ˙ γ ( − i ¯ σ γβ σ β ˙ α ) (cid:17) . (29)Then we define the Nambu-Gor’kov basis asΨ := 1 √ (cid:16) q Lα , q R ˙ α , ¯ q cRβ , ¯ q c ˙ βL (cid:17) T , ¯Ψ := 1 √ (cid:16) ¯ q αR , ¯ q L ˙ α , q c βL , q cR ˙ β (cid:17) . (30)The advantage of the Nambu-Gor’kov formalism is that we can naturally include the con-densates in the propagator of Ψ. For example, the diquark condensate in the s -wave ( LL or RR condensate) is given as ψ T C T γ ψ = − iq cL q L + iq cR q R , (31)thus the inverse propagator that contains this condensate can be written as G ( p ) − = − i p ν − µδ ν, ) σ ν i ¯∆ R ( p ) 0( p ν − µδ ν, )¯ σ ν − i ¯∆ L ( p ) i ∆ L ( p ) 0 0 ( p ν + µδ ν, ) σ ν − i ∆ R ( p ) ( p ν + µδ ν, )¯ σ ν . (32)The ∆-matrices appearing in the inverse propagator are defined as∆ L,R ( p ) = ∆ + ( p ) P L,R + ( p ) + ∆ − ( p ) P L,R − ( p ) , (33)where ∆ ± ( p ) are the gaps and the quark (anti-quark) on-shell projectors, P L,R ± ( p ), aredefined in Appendix A. The ones with the bar can be obtained by replacing P → ¯ P in theabove expression. (We assume the gaps to be real.) We can invert the matrix (32) by usingthe formulas listed in Appendix A. If we write G ( p ) = (cid:18) G G G G (cid:19) , (34) Note that the projectors are not invertible, so one must use appropriate inversion formulas.
15e then have G = − ∆ + P L − p − ( | ~p |− µ ) − ∆ − ∆ − P L + p − ( | ~p | + µ ) − ∆ − ∆ + P R − p − ( | ~p |− µ ) − ∆ + ∆ − P R + p − ( | ~p | + µ ) − ∆ − , (35)and other components will not be important in writing down the Dyson-Schwinger equa-tions. Notice that if the condensates, ∆ ± , in the denominators vanish there are terms thatdiverge as the energy is scaled toward the Fermi surface, i.e. , p → | ~p | → µ . This isessentially due to the sign structure of the chemical potential in the matrix (32) and to thefact that the ∆-matrices occupy off block-diagonal components. This infrared divergence iscured by the formation of the condensate and the ∆ properly behaves as such a condensate.If this were the traditional four-dimensional set up, we could have introduced the p -wavediquark condensate ( LR -pair) in the Nambu-Gor’kov propagator. [Such condensate wouldhave occupied the anti-diagonal slots of the matrix (32).] However, this is not allowed inour theory. As shown in Figure 9, since q L and q R separately live on the D8 and D8 branes, D8D8 D8 D8 D80,12,3 4
Allowed Not Allowd
Figure 9:
The left diagram represents the right-hand side of the Dyson-Schwinger equation (Figure 8) for LL -diquark condensate. On the right panel is the similar diagram with LR -diquark condensate. This clearlyis not making sense, because the condensate (the triangle) is separated and also is not being able to flip thehelicity. the Dyson-Schwinger equation for such condensate cannot make sense. In the previousmacroscopic renormalization group analysis, we encoded all the information about the extradimension in the form factors and the left- and right-handed quarks effectively lived in thesame four dimensional spacetime. However, in this microscopic Dyson-Schwinger analysis,we see that it is actually not possible for the LR -condensate to form. We emphasize thatthe condensate is not being energetically suppressed but simply not possible to form in thebrane system of Sakai and Sugimoto. For the particle-hole ( L ¯ R or ¯ LR ) pair, we might naively introduce the condensate inthe diagonal slots of the matrix in (32). However, this represents the introduction of theusual chiral condensate, that is, the pair of particle and anti-particle, and not the desiredparticle-hole pair. One can verify in this case that the infrared divergence near the Fermi Refs. [29, 30] discuss the condensation of a gauge invariant particle-antiparticle pair across the D8 andD8 branes at strong coupling. Although it is possible that this sort of condensation forms in our situation,we note that we are considering the condensation of a particle pair which is not gauge invariant and at weakcoupling. Therefore, this possibility is not taken into account in our discussion here. √ (cid:0) q Lα , q R ˙ α , q Lα , q R ˙ α (cid:1) T , (36)and consider the inverse propagator of the form (32) with the replacements, i ∆ L → Σ L , − i ∆ R → Σ R , − i ¯∆ L → ¯Σ L and i ¯∆ R → ¯Σ R , where the Σ-matrices are similarly defined as forthe ∆-matrices. The spinors of the second set have the chemical potentials in their kineticterms with opposite sign from the first set. This effectively introduces the hole degrees offreedom. In this case, the propagator has the infrared divergence near the Fermi surfaceand the Σ-condensate provides the cutoff. Thus we properly have the interpretation thatthe condensate is the particle-hole pair near the Fermi surface. We note that since thecondensate is formed out of the spinors with the dotted- or undotted-index pair, and notthe mixed one, the condensate lives either on D8 or D8 branes and not across them.We must also consider the gluon propagator which is the other ingredient of the Dyson-Schwinger equation. Unlike previous macroscopic treatment, we properly take into accountof the perturbatively computable screening effect. The most general form of the O (3)-invariant gluon propagator is D µν ( p, n ) = P Tµν ( p ) p + (2 πn/L ) + G s ( p ) + P Lµν ( p ) p + (2 πn/L ) + F s ( p ) , (37)where F s and G s are the electric and magnetic screenings, respectively, and the projectorsare defined as P Tij ( p ) = η ij − p i p j | ~p | , P T ( p ) = 0 = P T ( p ) , P Lµν ( p ) = η µν − p µ p ν p − P Tµν . (38)We have dropped the term with the gauge fixing parameter in the propagator. This termhas been verified not to contribute to the gap, the solution to the Dyson-Schwinger equation,at very large chemical potential [31].As explained in Section 2.1, the finite density screening effect is given by the diagramshown in Figure 1, at one-loop level. Since the quark loop stays on the D8 or D8 branes,there is no extra dimensional effect on the screening, and the gluons with nonzero momentumin the x -direction, such as the case shown in Figure 9, do not have the screening. We thushave the standard expressions [32] F s ( p ) = 2 m D p | ~p | (cid:20) − p | ~p | Q (cid:18) p | ~p | (cid:19)(cid:21) δ n, ,G s ( p ) = m D p | ~p | "( − (cid:18) p | ~p | (cid:19) ) Q (cid:18) p | ~p | (cid:19) + p | ~p | δ n, ,Q ( x ) = 12 ln (cid:12)(cid:12)(cid:12)(cid:12) x − x (cid:12)(cid:12)(cid:12)(cid:12) − i π θ (1 − x ) , m D = 14 π N f g µ , (39)where θ is the Heaviside function and δ n, signifies that the screening is effective only forthe gluons with n = 0. 17e can now write down the Dyson-Schwinger equation. We start with the equation forthe ∆-condensate. At high density (weak coupling), we can approximate the quark-gluonvertex with the bare ones and we have G ( k ) − − G ( k ) − = − ig X ( D ) X n Z d p (2 π ) Γ µ G ( p )Γ ν D µν ( q, n ) . (40)We have included the color factor X ( D ) as in Equations (9) and the representation D iseither symmetric or antisymmetric depending on the channel that G is in. Also we havedefined G as G without the condensates, q := k − p , and Γ µ areΓ µ := σ µ σ µ − σ µ − ¯ σ µ . (42)We can use (32), (35) and (40) to obtain the gap equation∆ ± ( k ) = ig X ( D ) X n Z d p (2 π ) tr (cid:20) σ µ (cid:18) ∆ + ( p ) P R − ( p ) p − ( | ~p | − µ ) − ∆ + ∆ − ( p ) P R + ( p ) p − ( | ~p | + µ ) − ∆ − (cid:19) ¯ σ ν P L ± ( k ) (cid:21) D µν ( q, n ) , (43)where the trace is over the spinor indices and is taken to project out ∆ ± from the ∆ L -matrixin Equation (33). After some algebra, one obtains∆ + ( k ) ≈ − ig X ( D ) X n Z d p (2 π ) (cid:20) ∆ + ( p ) p − ( | ~p | − µ ) − ∆ + ( p ) (cid:18) − (ˆ p · ˆ q )(ˆ k · ˆ q ) q + (2 πn/L ) + G s ( q ) + + ˆ p · ˆ kq + (2 πn/L ) + F s ( q ) (cid:19) + ∆ − ( p ) p − ( | ~p | + µ ) − ∆ − ( p ) (cid:18) p · ˆ q )(ˆ k · ˆ q ) q + (2 πn/L ) + G s ( q ) + − ˆ p · ˆ kq + (2 πn/L ) + F s ( q ) (cid:19)(cid:21) , (44)where we have assumed that the gaps are functions only of k or p . In deriving thisequation, we have adopted the approximation, q ≪ | ~q | ≈ µ , so that P Lµν ≈ η µν δ µ, δ ν, .The equation for ∆ − ( k ) is the same except that the two terms in the round brackets areexchanged. Note that only the first term in the equation of ∆ + ( k ) has the near-Fermi-surface (infrared) divergence that is being cured by the formation of the condensate. Thusto the first approximation in large µ , we can neglect the second term. Similar observationin the equation of ∆ − ( k ) results in the conclusion that this gap does not form at near theFermi surface. The easiest way to see how this color factor comes in is to note that X a T αab T αcd = 12 X (symm)( δ ab δ cd + δ ad δ bc ) + 12 X (asymm)( δ ab δ cd − δ ad δ bc ) . (41) ~p = ~p F + ~l k , and approximate | ~p F | ≈ µ , ˆ p F · ˆ l k ≈ | ~p | ≈ µ + l k and q ≈ µ (1 − ˆ p · ˆ k ). Then the integration measure takes the form µ dp dl k d cos θdφ withcos θ := ˆ p · ˆ k . Since the integral is dominated by the region θ ≈
0, we further approximatethat ˆ p · ˆ k ≈ p · ˆ q ≈ ≈ ˆ k · ˆ q in the numerators of the integral, but not in thedenominators. We Wick rotate p → ip and integrate over l k and φ . The gap equationnow takes the form∆ + ( k ) ≈ − g π X ( D ) X n Z dp d cos θ (cid:18) − cos θ + (1 / { πn/ ( µL ) } + G s ( q ) / (2 µ )+ 11 − cos θ + (1 / { πn/ ( µL ) } + F s ( q ) / (2 µ ) (cid:19) ∆ + ( p ) p p + ∆ + ( p ) , (45)with the approximate form of the screenings F s ( q ) ≈ m D δ n, , G s ( q ) ≈ π m D q | ~q | δ n, . (46)From this equation, it is clear that for the symmetric channel D = symm, we only have thetrivial solution ∆ + = 0. We thus consider the antisymmetric channel from now on.In Equation (45), the sum over n and integral over θ can be carried out in a straight-forward manner and yields∆ + ( k ) ≈ g π N c + 12 N c Z dp ln (cid:18) Λ | k − p | (cid:19) ∆ + ( p ) p p + ∆ + ( p ) , (47)where we have defined Λ := 2 √ π N − / f g − µ { sinh( µL ) / ( µL ) } , (48)and the part, {· · · } , is the contribution from the sum over n = 0. The factor | k − p | appearing in the logarithm comes from the Landau damping of G s ( q ) as in (46) and thiseffect was first discussed by Son [28]. We can follow Appendix B of Reference [28] to solvethis equation and obtain∆ + ( k ) = ∆ sin s π g N c N c + 1 ln Λ k ! , with ∆ = Λ exp " − π s π g N c N c + 1 . (49)Let us comment on this result. We first note that the gap vanishes in the ’t Hooftlimit. This is consistent with what we have concluded in the renormalization group anal-ysis. Now, when µL ≪
1, we have sinh( µL ) /µL ≈
1, so the extra dimensional effect inΛ disappears and the resulting expression for the gap coincide with the QCD result for N c = 3. (See, for example, Reference [1].) In the opposite limit where µL → ∞ , we havesinh( µL ) → exp( µL ) /
2, so the extra dimensional effect contributes heavily and the gapgrows with the parameter µL . As one can observe in Equation (45), this is because theinfrared effect of the terms with n = 0 comes to be comparable to that of the n = 0 term. If the flavor structure is included, this conclusion gets slightly more complicated. However, the factthat the antisymmetric channel dominates over the symmetric one does not change. See, for instance,Reference [33]. µL ∼ /g , the gap is much larger than the size of theFermi sphere itself and such a solution cannot be accepted, for the dynamics is no longertaking place near the Fermi surface. We therefore conclude that when µL ∼ /g , the gapdoes not form and the ground state simply is described by the Fermi liquid.We now turn to the Σ-condensate, χ DW. The computation is almost identical to theprevious case and we arrive at the equation similar to Equation (45) with the replacements∆ + → Σ + and D → • . There are, however, a few differences. The most important one isthe range of the integration parameter θ . This is restricted to the near infrared region, i.e. , θ ≈
0, because this is a forward scattering and the exchanged gluon should not be harderthan the size of the gap or the momentum carried by the propagator G ( k ). When the angleis set to small value, the l k component is about µ (1 − cos θ ), and the propagator carries themomentum approximately q p + Σ . We thus require µ (1 − cos θ ) ≤ q p + Σ , (50)and this inequality sets the upper limit on θ . Now because of this kinematic restriction, when N c ≈
3, the ∆-condensate dominates over the Σ-condensate. Therefore, in the following,we consider the ’t Hooft limit (with small λ ). In this limit, the screening F s and G s is1 /N c -suppressed, so we drop the screening terms from the gap equation. In the absence ofthe screening, the approximation, q ≈ µ (1 − cos θ ), has infrared problem when n = 0.This means that q cannot be neglected in this case and we must use q ≈ µ (1 − cos θ ) + | k − p | δ n, . We thus have the gap equation for the Σ-condensateΣ + ( k ) = g π N c − N c X n Z dp d cos θ − cos θ + | k − p | / (2 µ ) δ n, + (1 / { πn/ ( µL ) } Σ + ( p ) p p + Σ + ( p ) , (51)where the integration range of θ is restricted as mentioned above.Let us first consider the case with n = 0. In this case, we can carry out the cos θ integralwith the restriction (50) and obtainΣ + ( k ) ≈ g π N c − N c Z dp ln µ p p + Σ + ( p ) | k − p | ! Σ + ( p ) p p + Σ + ( p ) . (52)We can again solve this equation following Son [28] (also see [19]). In this case, the cal-culation is slightly different from Reference [28], so it is shown in Appendix B. The resultis Σ + ( k ) = Σ cos s g π N c − N c ln 2 µk , with Σ = 2 µ exp " − π s π g N c N c − . (53)This result agrees with Reference [13].When we include the terms with n = 0, the logarithm term in Equation (52) getsaugmented as ln 2 µ p p + Σ + ( p ) | k − p | + 4 ln sinh( µLǫ / ) µLǫ / , (54)20here ǫ := q p + Σ /µ . If µL is not too large compared to ǫ / , then the second term inthe above expression is small and the result (53) does not change. However, when µL is solarge that the second term yields dominant contribution ∼ µLǫ / , the integrand of the gapequation (52) takes the from proportional to 1 / ( p + Σ ) / , which is free of the infrareddivergence even without the condensate Σ. This implies that the gap does not exist. Toroughly estimate the value of µL at which the crossover occurs, we approximate the firstterm in Equation (54) as ln(2 µ/ Σ ) and the second term as µL (Σ /µ ) / . Then we see thatthe second term becomes important when µL & ( µ/ Σ ) / ln( µ/ Σ ) ≈ e / √ λ / √ λ . (55) We have addressed the possibilities of the color superconductivity and the chiral densitywaves in the Sakai-Sugimoto model at finite density and explicitly carried out the compu-tations in the weak coupling limit. As was stated in the introduction, the ultimate goalof this investigation is to obtain the quantitative behavior of the model in the weak andstrong coupling regions, make qualitative comparison of the phase diagrams and gain insightinto the QCD phase diagram which is still unsettled. We therefore comment on the strongcoupling gravity background analysis of the model at finite density.As suggested by Sakai and Sugimoto [15], the strong coupling analysis is done by tak-ing the gravity background limit of the D4-branes while treating the D8-branes as probes,then the DBI-action of the probes are studied to obtain the spectrum of the low energyexcitations at strong coupling. The generalization to the finite density has been discussedin References [34, 35, 36]. The phase diagram of the model in the space of temperatureand chemical potential has been obtained by Horigome and Tanii [35]. Let us briefly reviewtheir results. In the previous section, we have placed the D8- and D8-branes at the an-tipodal points of the compactified circle. However, this is not necessary and in this gravitybackground analysis, the compactification radius is set to R and the distance between thebranes to L with the range 0 < L ≤ πR .At finite temperature, in addition to the x -direction, the Euclidean time direction, τ , isalso compactified and the period of the time circle is identified with the inverse temperature.Horigome and Tanii consider the three known phases, first discovered by Aharony et al. [37]at zero density, and the spacetime configurations for the phases are illustrated in Figure 10.In the figure, the vertical U -axis represents the radial direction in 5,6,7,8 and 9 directions.The change in the background geometry of the phases from (a) to (b,c) is interpreted asthe confinement/deconfinement phase transition by Aharony et al . The thinner lines andcurves in the diagrams represent the D8 or D8 branes and the change in the configurationfrom Diagram (b) to (c) is interpreted as the chiral symmetry restoration.The on-shell DBI actions of the D8-branes with nonzero chemical potential have beenobtained by Horigome and Tanii for each phase. For the configurations with the smooth U-shaped D8-branes, that is, for the diagrams (a) and (b) of Figure 10, they have foundthat the chemical potential must be constant along the radial U -direction and the actionsfor those configurations are independent of the chemical potential. Only for the parallel D8-brane configuration of Diagram (c) has the non-trivial dependence on the chemical potentialin its action. By comparing the on-shell actions at the various values of the temperature and21 SB χ x SB χ x x U τ confined(a) τ (b) deconfined τ (c) deconfined χ Figure 10:
The spacetime configurations of the three phases. The vertical axis U is the radial directionin 5,6,7,8 and 9 directions. The thinner lines represent the stacks of D8 and D8 branes. Diagram (a) isthe low temperature confined phase. The chiral symmetry is broken in this phase. Diagram (b) is the hightemperature deconfined phase and with χ SB. Diagram (c) is also the high temperature deconfined phasebut with chiral symmetry restored. the chemical potential, they determined the phase diagram which is schematically shown inFigure 11. (b) c SB χ S χ T µ ConfinedDeconfined T c SB χ S χ SB χ T µ ConfinedDeconfinedDeconfined
L > 0.97 R L < 0.97 R (a) T Figure 11:
The schematic phase diagram obtained by Horigome and Tanii. The temperature T c denotesthe confinement/deconfinement phase transition temperature. The inter-D8D8 distance L ≃ . R is thecritical value where the deconfined phase with χ SB exists.
The confinement/deconfinement phase transition line at T = T c is determined by theD4-background geometry and the D4 difference action in those phases scales as N c [37].This is expected to completely dominate the D8 probe actions, which scale as N c N f , sothe confinement/deconfinement phase transition line is not affected by the value of thechemical potential. Therefore, Horigome and Tanii assume that the phase below theconfinement/deconfinement line is in the configuration (a) of Figure 10, and compare theD8 actions of the configurations (b) and (c) in the deconfined D4-background geometryto obtain the phase structure above the confinement/deconfinement phase transition line.This is why the χ S/ χ SB phase transition line of the right panel in Figure 11 is terminatedat T = T c . When the inter-D8D8 distance L is larger than 0 . R , it has been shown byAharony et al. [37] that the phase represented in Diagram (b) of Figure 10 does not exist.Therefore, the phase diagram becomes rather structureless, as shown in the left panel ofFigure 11. In Reference [35], the temperature and chemical potential are measured in different units. If measuredin the same units, the chemical potential is asymptotically larger than the temperature by the factor of the’t Hooft coupling λ . In the original version of this e-print, we overlooked the dominance of the D4 action and compared onlythe D8 actions in the confined and deconfined phases, which was incorrect.
QCD (at low temperature), the hadrons start to overlap and the quarks are shared bymany hadrons. This implies the change in the degrees of freedom and since only the twophases are assumed, there should be a phase transition. Stated differently, in this simplifiedpicture of QCD, we expect that a phase transition to occur even at zero temperature. SeeFigure 12. Though the physical setups are similar, we see that the phase diagrams in µ T Qaurk Gluon PlasmaHadron
Figure 12:
Expected phase diagram of the simplified QCD where only the hadronic and quark-gluonplasma phases are assumed to exist.
Figures 11 and 12 are qualitatively different. We suspect that the discrepancy stems fromthe large N c limit and the probe approximation in the Sakai-Sugimoto model. As we haveexplained, the confinement/deconfinement phase transition is determined by the D4 actionsbecause the difference action scales with N c and dominates over the D8 actions (with thechemical potential). It is unlikely that this picture can change in the probe approximation.Therefore, in order to see modifications in the confinement/deconfinement phase transitionline, one should take into account of the D8-brane back reaction to the geometry.It is also clear that we do not observe the color superconductivity or chiral density wavesin the strong coupling analysis of Horigome and Tanii because these possibilities are simplynot considered. To explore those exotic phases, one must come up with the correspondingstringy pictures of the branes and strings, and see if they are energetically preferred at anypoint in the µ - T parameter space. Such stringy pictures of quark matter are, so far, notclear to us. Nevertheless, we make some general remarks in this direction. First, we mustremember that the relevant degrees of freedom at high density are particles and holes whilethe anti-particles are buried deep in the Dirac sea. Thus, if the U-shape configuration ofthe D8-branes describes the mesons, which are the pairs of particle and anti-particle, thenwe expect the configuration of the superconductor or chiral density waves to be differentfrom the U-shape configuration because they are the pairs of particles and holes. Secondly,the Fermi sphere plays the essential role in high density QCD. Thus, the Fermi sphere must It is interesting to observe that the weakly coupled large N c N = 4 super-Yang-Mills theory on threesphere and Type IIB supergravity on AdS × S both have qualitatively the same phase structure as Figure 12[38, 39, 40]. In these cases, however, the chemical potential is conjugate to the U (1) subgroup of SU (4) R -symmetry. The stringy configurations of baryons have been suggested in Refs. [41, 42] in which the nuclear matterphase has been discovered in the phase diagram.
23e encoded in the holographic picture somehow. Also, the holography is in the ’t Hooftlimit under which we do not expect non-gauge invariant quantities to survive. Therefore,it is likely that we do not observe the color superconducting phase and the relevant phaseprobably is the chiral density waves in the holographic theories.Although Shuster and Son [14] settled that the color superconductivity dominates overthe chiral density waves in the high density weakly coupled QCD (with N c = 3), it still isnot known if this observation persists at the medium density strong coupling region. TheSakai-Sugimoto model in the usual ’t Hooft limit is not likely to be able to address thisproblem but it is interesting to examine this in the limit with N f /N c fixed. In this case, wecan expect the effect of the color-flavor locking to be significant and also the competitionbetween the D4 and D8 actions becomes non-trivial.Finally, as we have mentioned in the introduction, the rich structure of the QCD phasediagram is partly due to the quark masses. As the authors of Reference [21] have alreadynoted, the inclusion of the quark masses involves the tachyon that comes from the stringstretching between the D8 and D8 branes. This subject is being actively studied (forexample, see Refs.[23, 24, 25, 29, 30]) and the resulting phase diagram is yet to be seen. If the finite density holographic model turns out to capture the aspects of QCD, it wouldbe very interesting because we can explore the region of the QCD phase diagram where theperturbative nor the numerical analysis is available. Acknowledgments
I would like to thank the members of Racah Institute of Physics who provided valuablequestions and comments. I benefited from Oren Bergman and Jacob Sonnenschein withregard to the strongly coupled aspects of the theory. I also would like to thank AndreiKryjevski for educating me high density QCD in year 2004. This work was supported bythe Golda Meir Post-Doctoral fellowship.
A Some Formulas
We adopt the convention of Wess and Bagger [26], except the definition of the Dirac spinoras in Equation (5). Many useful formulas can be found in Appendix B of the reference.The propagators of the left- and right-handed quarks are respectively given as − i ( p ν − µδ ν, ) σ ν α ˙ α ( p λ − µδ λ, ) , and − i ( p ν − µδ ν, )¯ σ ν ˙ αα ( p λ − µδ λ, ) . (56)We define the charge conjugation matrix C := (cid:18) iσ ¯ σ i ¯ σ σ (cid:19) , (57)which satisfies C − γ ν C = − γ νT and C − = C T = − C .24he quark and anti-quark on-shell projectors are defined as P L ± ( p ) := 12 (cid:0) ± σ ¯ σ j ˆ p j (cid:1) , P R ± ( p ) := 12 (cid:0) ± ¯ σ σ j ˆ p j (cid:1) , ¯ P L ± ( p ) := 12 (cid:0) ± ˆ p j ¯ σ j σ o (cid:1) , ¯ P R ± ( p ) := 12 (cid:0) ± ˆ p j σ j ¯ σ (cid:1) , (58)where ˆ p := ~p/ | ~p | . Notice that P L ± = ¯ P R ∓ and P R ± = ¯ P L ∓ .We list the useful formulas for inverting the inverse of the Nambu-Gor’kov propagator.For n × n matrices A , B , C and D , we have (cid:18) A BC D (cid:19) − = (cid:18) A − + A − BS − A CA − − A − BS − A − S − A CA − S − A (cid:19) , (59)provided that the matrices A and S A := D − CA − B are invertible. We also have (cid:18) A BC D (cid:19) − = (cid:18) − C − DS − C C − + C − DS − C AC − S − C − S − C AC − (cid:19) , (60)provided that C and S C := B − AC − D are invertible. Other convenient formulas are thefollowing. ( p ν − µδ ν, ) σ ν = ( p + | ~p | − µ ) σ P R + + ( p − | ~p | − µ ) σ P R − , ( p ν − µδ ν, )¯ σ ν = ( p + | ~p | − µ )¯ σ P L + + ( p − | ~p | − µ )¯ σ P L − . (61) { ( p ν − µδ ν, ) σ ν } − = ¯ σ P L + p − | ~p | − µ + ¯ σ P L − p + | ~p | − µ , { ( p ν − µδ ν, )¯ σ ν } − = σ P R + p − | ~p | − µ + σ P R − p + | ~p | − µ . (62) P L ± σ P R ± = 0 , P L ± σ P R ∓ = σ P R ∓ ,P R ± ¯ σ P L ± = 0 , P R ± ¯ σ P L ∓ = ¯ σ P L ∓ . (63) B Solving Gap Equation
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