A holographic model for QCD in the Veneziano limit at finite temperature and density
T. Alho, M. Jarvinen, K. Kajantie, E. Kiritsis, C. Rosen, K. Tuominen
PPrepared for submission to JHEP
CCTP-2013-19CCQCN-2013-6HIP-2013-20/THCERN-PH-TH//2013-320
A holographic model for QCD in the
Veneziano limit at finite temperature and density
T. Alho a,b
M.J¨arvinen c K. Kajantie b E. Kiritsis c,d,e
C. Rosen c K. Tuominen f,ba
Department of Physics, P.O.Box 35, FI-40014 University of Jyv¨askyl¨a, Finland b Helsinki Institute of Physics, P.O.Box 64, FI-00014 University of Helsinki, Finland c Crete Center for Theoretical Physics, Department of Physics, University of Crete, 71003Heraklion, Greece. d APC, Univ Paris Diderot, Sorbonne Paris Cit´e, UMR 7164 CNRS, F-75205 Paris, France. e Theory Group, Physics Department, CERN, CH-1211, Geneva 23, Switzerland f Department of Physics, P.O.Box 64, FI-00014 University of Helsinki, Finland
E-mail: [email protected] , [email protected] , [email protected] , hep.physics.uoc.gr/ ∼ kiritsis/ , [email protected] , [email protected] Abstract:
A holographic model of QCD in the limit of large number of colors, N c ,and massless fermion flavors, N f , but constant ratio x f = N f /N c is analyzed at fi-nite temperature and chemical potential. The five dimensional gravity model containsthree bulk fields: a scalar dilaton sourcing Tr F , a scalar tachyon dual to ¯ qq and a4-vector dual to the baryon current ¯ qγ µ q . The main result is the µ, T phase diagramof the holographic theory. A first order deconfining transition along T h ( µ ) and a chiraltransition at T χ ( µ ) > T h ( µ ) are found. The chiral transition is of second order forall µ . The dependence of thermodynamical quantities including the speed of soundand susceptibilities on the chemical potential and temperature is computed. A newquantum critical regime is found at zero temperature and finite chemical potential. Itis controlled by an AdS × R geometry and displays semi-local criticality. Keywords:
Gauge/ gravity duality, holography, QCD phase diagrams a r X i v : . [ h e p - ph ] O c t ontents and the DBI Black-Hole Solution 133.2 AdS Solution 153.3 A closer look at the AdS solution 173.4 Stability of the AdS Region 19 λ h , ˜ n plane 274.5 Constant parameter curves 30 T → point 42 – i – Numerical solution of the equations of motion 45
B.1 Definitions 45B.2 The f -scaling 46B.3 The Λ-scaling 47B.4 Physical observables at µ = 0 49B.5 Chemical potential 50B.6 Physical observables for µ (cid:54) = 0 53 C Determination of τ h ( λ h ; m q )
53D Numerical results for T and µ
56E Thermodynamics along λ end
56F Large scale behavior 59
The phase diagram of QCD, as a function of temperature T and chemical potential µ ,corresponding to baryon density or some other conserved charge like isospin, displays arich structure [1]. Particularly interesting and important features of the phase diagramare the nature of the chiral phase transition, the location of the chiral critical pointand its properties. All these have been extensively studied both with effective chiralmodels [2, 3] and other approaches reviewed e.g. in [4], and holography [5–7]. Sincefirst principle lattice methods [8–10] are currently still limited to small values of µ/T ,the model studies provide important complement. However, the location of the criticalpoint is very dependent on the details of the models [4].In addition to temperature and density, typically also external perturbations of thechiral symmetry, i.e. finite quark masses, are present and provide further dimensionsto the phase diagrams. For example, in the limit of vanishing quark masses, the finitetemperature phase transition of two flavor QCD at zero chemical potential is expectedto be of second order. At low temperatures, the chiral transition at finite µ is expectedto be of first order [2]. At intermediate temperatures and densities the first and secondorder transition lines were conjectured to meet at a tricritical point. The finite quarkmass softens the singularity at the second order phase transition which becomes asmooth crossover. The line of first order transitions is unaffected in the presence ofsmall external perturbation, and the tricritical point becomes the critical endpoint for– 1 –his line. The fate of the existence of the critical point of QCD in the ( µ, T )-plane atthe physical value of quark masses is ultimately determined by the form of the criticalmanifold in the multidimensional space of parameters µ, T, m q [11]. One can imagineseveral possibilities to occur. Indeed, it can be that the existence of a critical pointnear the chiral limit implies that the critical point does not exist at physical masses.Depending on the shape of the critical manifold, a variety of other possibilities can beimagined.Effective field theories utilizing holographic methods, motivated by the AdS/CFTcorrespondence [12–14], have become a major tool in the analysis of strongly coupledtheories both in elementary particle and condensed matter physics [15, 16]. A classof bottom up models for QCD-like theories, which captures the entire renormalizationgroup evolution of the corresponding quantum field theory from weak to strong cou-pling has been developed in [17–19]. A particular application of this framework is thedetermination of the vacuum and finite temperature phase diagrams of the associatedquantum gauge theories [18, 20–26]. The framework has been extended to account forthe dynamics of chiral symmetry breaking in the presence of flavors [27–32]. In orderto consider effects coming from the backreaction of flavor to color, holographic modelswith dynamics close to that of QCD in the Veneziano limit were explored and developed[32–35]. In this work we consider adding finite chemical potential in order to deter-mine the phase diagram in the ( T, µ )-plane by computing the pressure p ( T, µ ; m q = 0)in the phases where chiral symmetry is intact or spontaneously broken.Concretely, we consider the holographic model for equilibrium QCD with N f mass-less quarks at the limit N f → ∞ , N c → ∞ and fixed ratio x f = N f /N c . For a thoroughdiscussion of the fundamentals of this type of bottom-up holographic model for QCDin the Veneziano limit (V-QCD) at zero or finite T but zero density, we refer to [33, 34].Here we only outline the features arising when we allow also finite density and chemicalpotential in V-QCD. According to the holographic dictums to add baryon density wemust turn on a source for the five-dimensional gauge field A a . The dynamics of thebaryon number gauge field A a is determined by its appearance in the tachyon DBIaction, which can be schematically written as (cid:112) − det ( g ab + κ ∂ a τ ∂ b τ + w F ab ) . (1.1)Here κ and w are couplings, F ab = ∂ a A b − ∂ b A a , and τ is the tachyon, sourcing ¯ qq . Toturn on a uniform constant density, the Ansatz A a = Φ( z ) δ a should be made, where z is the coordinate of the 5th dimension and the only non-zero component of F ab is F z = ∂ z Φ( z ). The action contains only the derivative of Φ and the finite density arises For a different effort in that direction, see [36]. – 2 –s the integration constant ˜ n of the equation of motion of the cyclic configuration spacecoordinate Φ.The three bulk fields λ, Φ , τ correspond to the three arguments in p ( T, µ ; m q ), andwe will consider only the case m q = 0 in this paper and denote the pressure simplyby p ( T, µ ). As in [34] we find that there are two types of m q = 0 solutions: thosewith vanishing tachyon (chirally symmetric) and those with nonzero tachyon (breakingchiral symmetry spontaneously). To determine the pressure, the strategy is thereforeto find black hole solutions with one or two scalar hair (corresponding to the dilatonand tachyon scalars) and a non-trivial charge density. Such solutions, when they ex-ist, compete also with finite temperature but zero charge solutions without a blackhole. The reason is that these zero charge solutions always have a constant Φ = µ andtherefore correspond to saddle points with finite chemical potential but zero chargedensity. Such solutions are expected to dominate at small enough temperature andchemical potential, and we identify them with the “hadron gas” vacuum phase withzero pressure. Increasing the charge density, we have the possibility of a trivial ornon-trivial tachyon field. The latter possibility describes a “deconfined” but chiralitybreaking plasma, while the former corresponds to chirally symmetric plasma. To de-termine which of these two dominates, one solves numerically for the coupled equationsof motion of the fields, and finds pressures p s ( T, µ ) and p b ( T, µ ) corresponding, respec-tively, to the solutions with intact or spontaneously broken chiral symmetry. Equalityof pressures, temperatures and chemical potentials then defines the phase boundary onthe
T, µ plane.The main outcome of this work is the phase diagram shown in Fig. 1 which wasobtained for the theory with x f = N f N f = 1, namely for the same number of masslessflavors and colors. The chiral transition is of second order for all µ , with the transitionline ending at zero temperature, µ ≈ .
6. For larger µ , the system is always in thechirally symmetric deconfined phase. There is also a tentative deconfining transitionat T h ( µ ) between the chirality breaking plasma and the “hadron gas” phase discussedabove. This phase boundary is determined by the condition p b ( T, µ ) = p low = 0. To motivate this in the field theory, note that the degrees of freedom of the lowtemperature phase are the Goldstone bosons of the spontaneously broken chiral sym-metry, and their number is ∼ N f . On the other hand, the number of degrees of freedomin the high-temperature phase is ∼ N c + N c N f . As we consider only the case x f = 1 It is well known that in the presence of flavor there is no order parameter for deconfinement:confined phases can be continuously connected to Coulomb and Higgs phases. However, at large N c the pressure itself can be considered as an order parameter for deconfinement. The confined phasehas p ∼ O (1), while deconfined phase has p ∼ O ( N c ). When we talk about confined and deconfinedphases we have this definition in mind. – 3 –e obtain p low /p high ∼ / ∼
0. The relative weight of the low-temperature degrees offreedom grows with x f , and ultimately at some x c (cid:39)
4, in terms of the free energy, theybecome indistinguishable from the high temperature ones. This signifies the quantumphase transition from a confining gauge theory to the one whose long-distance behaviorat zero temperature is governed by a nontrivial and stable infrared fixed point. Weleave the study of the finite temperature and density phases in the limit x f → Hadron gas Χ SB plasmaChirally symmetric plasma Μ T Figure 1 . Chemical potential dependence of transition temperatures of the deconfining( T h ( µ )) and chiral ( T χ ( µ )) transitions at m q = 0. The dashed line corresponds to a secondorder phase transition while the solid lines corresponds to a first order transition. If finitequark mass is turned on, the second order transitions become smooth crossovers. The T = 0lines in the χ SB plasma phase as well as the chirally symmetric phase correspond to a newquantum critical semilocal phase at finite density. Except for the lack of a critical point, all these features of this phase diagramagree on the general expectations. However, for the phase diagram of QCD at lowtemperatures there is a surprise: There exists a new quantum critical regime at T = 0,with exotic properties which realize the symmetries of the associated geometry, thatis AdS × R . The presence of the AdS × R geometry in the holographic solutionindicates that there is a scaling symmetry of the time direction which does not act inthe spatial directions. Such symmetries have been called semilocal. While this is anunexpected symmetry in a field theory at finite density, it is natural and generic inthe holographic context [37], and appears even in simple black holes as the Reissner-Nordstr¨om black hole [38]. This new scaling region exists on the T = 0 segment of thechirality breaking plasma as well as on the T = 0 line of the chirally symmetric plasma.– 4 –he physics in this critical regime is similar to that of a theory with zero speed of light:all spatial points decouple in the IR.It is well known that such AdS solutions are highly unstable as AdS has a ratherrestrictive Breitenlohner-Freedman bound. The instabilities associated to the fields weconsider can be understood in terms of the physics of the phase diagram and we describethem in detail in section 3.4. However, there can be further instabilities associated withother operators which we have not included here. It is possible that such quantumcritical points play an important role in the appearance of color superconductivity andcolor flavor locking at high density.There are many technical obstacles one has to cross before obtaining the finalnumerical results for the phase diagram: First, to find the relevant charged black-holesolutions one has to guarantee that the metric function f ( z ) vanishes at the horizon z = z h . As the horizon is a singular point of the equations, the numerical evolutionmust start close to the horizon with the appropriate boundary conditions. Second,the UV quark mass will be fixed to zero in order to have exact chiral symmetry. Toimplement this, we must solve the entire coupled set of equations of motion and tunethe boundary conditions so that the leading term of the tachyon field at small z (nearthe boundary) is ∼ z , instead of a linear one corresponding to a finite quark mass. Thisrequires high numerical precision in the solution of the non-linear equations of motion.The third difficulty is that the quantity to be computed is a function of two variables, p ( T, µ ). The numerics is correspondingly parametrised by two parameters, the value ofthe dilaton at the horizon λ h and the integration constant ˜ n . These parameters cannotbe continuous ones, but one can determine, say, T ( λ h , ˜ n ) as a function of λ h for fixedvalues of ˜ n , and vice versa. Proceeding in this way one obtains p ( T, µ ) on two gridson the
T, µ plane (see Fig. 21). Fourth and final issue is that one has to guarantee,using scaling properties of the equations of motion, that all the physically dimensionfulquantities are expressed in the same units.All of these considerations make the numerical problem at hand challenging. Inthis paper we focus on the details of introducing the chemical potential to the model,limit ourselves to one set of potentials chosen from [34] and to one value x f = 1. Thisallows us to show that the method works, produces interesting results and motivatesfurther studies. We have released the numerical code which has been used to computethe results presented in this paper [39].In Section 2 we specify the model and give the equations of motion and theirscaling properties. In Section 3 we find all solutions with constant scalars as they arecritical end points of flows. They correspond to AdS and AdS geometries and weanalyze their RG stability. In Section 4 we discuss the horizon expansion requiredfor initialising numerical solution and the physical values of the parameters λ h , ˜ n of– 5 –umerical integration. The main numerical results for the pressures, the transitiontemperatures T h ( µ ), T χ ( µ ) and sound velocity are shown and discussed in Section 5.1.The Appendices contain a detailed discussion of the numerical solutions, examplesof the computed values of T and µ and a detailed presentation of the chiral phasetransition line on the plane of numerical parameters λ h , ˜ n . The action of the model for vanishing chemical potential has been discussed thoroughlyin [33–35]. We focus here on the additional terms needed to describe the finite baryondensity. The action of the model is, in standard notation [34], S = 116 πG (cid:90) d x L, (2.1)where the Lagrangian is √− g (cid:20) R + (cid:2) − g µν ∂ µ φ∂ ν φ + V g ( λ ) (cid:3) − V f ( λ, τ ) (cid:112) − det [ g ab + κ ( λ ) ∂ a τ ∂ b τ + w ( λ ) F ab ] (cid:21) = b (cid:20) − fb (cid:32) bb + 4 ˙ b b + 8 ˙ bb ˙ ff + ¨ ff + 43 ˙ φ (cid:33) + V g ( λ ) − V f ( λ, τ ) (cid:114) f κb ˙ τ − w b ˙Φ (cid:21) . (2.2)The metric Ansatz isd s = b ( z ) (cid:20) − f ( z )d t + d x + d z f ( z ) (cid:21) , b ( z ) = e A ( z ) −→ z → L UV z , f (0) = 1 . (2.3)The functions b and f of the metric, the dilaton λ = e φ , the tachyon τ and the bulkdensity Φ depend only on the extra dimensional coordinate z . The Gibbons-Hawkingcounterterm is implied.The Lagrangian (2.2) is parametrized in terms of the potentials V g ( λ ), V f ( λ, τ ), κ ( λ )and w ( λ ) which are chosen to satisfy two basic requirements. First, in the ultraviolet,i.e. in the weak coupling limit, the model should reproduce the known perturbativebehaviors of the corresponding field theory. Second, in the deep infrared the modelshould lead to the generation of a dynamical wall shielding the singular behavior as λ → ∞ , which is responsible for confinement in the absence of the tachyon.In the numerical study in this article we will take the gauge coupling constantfunction w in the DBI action to be proportional to the other model function κ as w ( λ ) = L A κ ( λ ) . (2.4) This choice is at the boundary of allowed choices as indicated in [35]. – 6 –ere the scale L A ∼ L UV appears in order to match the dimensions correctly. It canbe formally eliminated from the Lagrangian (2.2) by rescaling Φ. In Appendix F weshall find that L A ≈ L UV ( x f = 0) = 1. Note that if one expands the Lagrangian (2.2)in F ab and writes it in the form − e F , one can identify a dimensionless coupling γ = e L UV2 = L UV2 V f w = L UV2 V f L A κ . (2.5)Explicitly, the potentials are [34] V g ( λ ) = 12 L (cid:20) λ
27 + 4619 λ (cid:112) λ )(1 + λ ) / (cid:21) , V f ( λ, τ ) = x f V f ( λ ) e − τ / L UV , (2.6)where the function V f ( λ ) is given by V f = 12 L x f (cid:20) L L − (cid:18) L L −
11 + 2 x f (cid:19) λ + 1729 (cid:18) L L − x f − x f (cid:19) λ (cid:21) ≡ W + W λ + W λ . (2.7)The scale L UV has a nontrivial dependence on x f , L UV = L (1 + x f ) / , which isdetermined by matching the pressure to the Stefan-Boltzmann limit at µ = 0 [34]. Thefunction κ ( λ ) is given by κ ( λ ) = [1 + ln(1 + λ )] ¯ µ [1 + ( − x f + ¯ µ ) λ ] / . (2.8)The numerical factors appearing in (2.6) and (2.7) simply provide the equivalencewith the known perturbative behavior in the weak coupling limit. This matching isobtained via the definition β ( λ ) = dλdb/b , (2.9)and recalling that the 2-loop beta function for the coupling λ = N c g ( µ ) / (8 π ) of theboundary theory is β ( λ ) = dλ ( µ ) d ln µ = − b λ − b λ , b = (11 − x f ) , b = (34 − x f ) (2.10)in the Veneziano limit. Analogously, the numerical factors appearing in κ ( λ ) in Eq.(2.8) are obtained by first defining γ ( λ ) = d ln τd ln b + 1 , (2.11)– 7 –nd then relating to the quark mass anomalous dimension with the scheme independentcoefficient γ defined by γ ( λ ) = d ln md ln µ = − γ λ + · · · , γ = 32 = 9 b − x f ) . (2.12)The actual numerical value of the quark mass (and the condensate (cid:104) ¯ qq (cid:105) ) is fixed by theUV expansion of the tachyon (remembering that the energy dimension of τ is − τ ( z ) / L UV = m q z ( − ln Λ z ) − γ b + (cid:104) ¯ qq (cid:105) z ( − ln Λ z ) γ b . (2.13)To have exact chiral symmetry one must find solutions for which m q = 0, and achievingthis, is one of the technically most demanding tasks of this model (for details, seeAppendix B).The behavior of the potentials at large values of the fields λ and τ is determined byrequirements of a confining spectrum and breaking of the chiral symmetry in the deepinfrared [18, 20, 22, 33]. To fix the last remaining parameter we choose ¯ µ = − . Thischoice, according to [34], leads to regular thermodynamics at zero chemical potential.With these definitions, the numerical results in this paper are given for the poten-tials (2.6)-(2.8) and for x f = 1 case only. Of course the above choice for the potentialsand κ is not unique but other possibilities exist as discussed in [33, 35]. The defini-tions presented above are taken in this paper to provide for a benchmark study of thismodel, and focused analyses of other potentials and other values of x f , in particularapproaching the conformal region at x f ≈
4, are left for future studies.As a final remark here, we emphasize that the duality between classical gravity andfield theory can be derived in the string theory framework only in the strong couplinglimit. In our case, the matching to the scheme independent perturbative results in theweak coupling limit has to be regarded as a model assumption, to be judged on thebasis of its consequences. Among these, an immediate and important one is that onecan describe thermodynamics up to arbitrarily high T and µ and identify solid knownbehaviors. Actually it is quite nontrivial that this matching can be carried out andthe correct running of the quark mass and the condensate implemented using the DBIaction. The model is then an effective theory extending weakly coupled results at large T, µ to the strongly coupled domain. Φ equation of motion The fermionic part of the action, given by L f [ τ, ˙ τ , ˙Φ] = V f ( λ, τ ) b (cid:114) f κb ˙ τ − w b ˙Φ , (2.14)– 8 –epends only on ˙Φ so that Φ is a cyclic coordinate. Since both L f and ˙Φ have energydimension 2, we have a dimensionless constant of integration ˆ n : ∂L f ∂ ˙Φ = − bV f w ˙Φ (cid:113) fκb ˙ τ − w b ˙Φ = ˆ n. (2.15)From this one solves˙Φ = − ˆ nb w (cid:115)(cid:18) f κb ˙ τ (cid:19) n + ( b wV f ) ≡ − ˆ nbV f w (cid:115)(cid:18) f κb ˙ τ (cid:19)
11 +
K , (2.16)where we have also introduced the dimensionless density factor K ( z ) = ˆ n b w V f = ˆ n L A b κ V f . (2.17)The factor K defined above contains the density effects in this holographic model andwill appear repeatedly in what follows.After the bulk fields λ and τ have been determined from their equations of motionand the Einstein’s equations, Φ( z ) can be computed by integrating Eq. (2.16):Φ( z ) = µ + (cid:90) z dz ˙Φ( z ) (2.18)with the constraint that the field Φ vanishes at the horizon z = z h ,Φ( z h ) = 0 = µ + (cid:90) z h dz ˙Φ( z ) , (2.19)from which µ is determined. Using the previous results for Φ, differential equations for b, λ, f, τ can be derived. Theyare for b ( z )3¨ bb + 6 ˙ b b + 3 ˙ bb ˙ ff − b f V g + b f V f (cid:18) f κb ˙ τ (cid:19) (cid:115) K fκb ˙ τ = 0 , (2.20)for λ ( z ) ¨ λλ − ˙ λ λ + 3 ˙ bb ˙ λλ + ˙ ff ˙ λλ + 38 b f λ ∂V g ∂λ −
38 1 √ K (cid:26) V f (cid:113) fκb ˙ τ λκ (cid:48) (cid:20) ˙ τ (1 − K ) − b f κ K (cid:21) + b f (cid:114) f κb ˙ τ λ∂ λ V f (cid:27) = 0 , (2.21)– 9 –or f ( z ) ¨ f + 3˙ bb ˙ f − ˆ n bw (cid:115) fκb ˙ τ ˆ n + b w V f = 0 , (2.22)and for τ ( z ) (1 + K ) ¨ τ − (cid:18) b f κ + ˙ τ (cid:19) ∂ ln V f ∂τ + f κ b (cid:20) d ln b f κdz + 2 ˙ λ ∂ ln V f ∂λ + K (cid:18) d ln( b f /κ ) dz (cid:19)(cid:21) ˙ τ + (cid:20) d ln b f κdz + ˙ λ ∂ ln V f ∂λ + K (cid:18) d ln fdz (cid:19)(cid:21) ˙ τ = 0 . (2.23)For λ we also have the first order equation12 ˙ b b + 3 ˙ bb ˙ ff −
43 ˙ λ λ = b f (cid:32) V g − V f (cid:115) K fκb ˙ τ (cid:33) . (2.24)It turns out useful to define the quantity V eff ( λ, τ ) = V g ( λ ) − V f ( λ, τ ) (cid:115) n b w V f . (2.25)Using this in τ = 0 case the equations can be written in more compact form as follows:First we have from the above definition V eff ( λ, τ = 0) = V g ( λ ) − (cid:115) V f ( λ,
0) + ˆ n b w ( λ ) . (2.26)Treating this as a function of λ and b , the three remaining equations of motion are then¨ bb − b b + 49 ˙ λ λ = 0 , (2.27)¨ λλ − ˙ λ λ + 3 ˙ bb ˙ λλ + ˙ ff ˙ λλ + 3 b f λ∂ λ V eff = 0 , (2.28)¨ f + 3˙ bb ˙ f − b ∂ b V eff = 0 . (2.29)The energy unit of the solutions is determined by fixing the small- z behavior of thedilaton to the perturbatively known field theory behavior, i.e. b λ ( z ) = − / ln(Λ z )with Λ = 1. To do this accurately enough, one has to go to extremely small values of– 10 – and it is better to use ln( z ) or actually ln b as the coordinate. For numerics we thuswrite the equations in the A = ln b basis changing z to A via the relation q ( A ) = e A dzdA , e − A dAdz = ˙ bb . (2.30)Then we have for q ( A ), primes denoting derivatives with respect to A ,12 − q (cid:48) q + 43 λ (cid:48) λ + 3 f (cid:48) f = q f (cid:32) V g − V f (cid:115) f κτ (cid:48) q √ K (cid:33) . (2.31)The remaining equations of motion are for λ ( A ) λ (cid:48)(cid:48) λ − λ (cid:48) λ + (cid:18) − q (cid:48) q (cid:19) λ (cid:48) λ + f (cid:48) f λ (cid:48) λ + 38 q f λ ∂V g ∂λ (2.32) − λ √ K (cid:26) (cid:113) fκq τ (cid:48) κ (cid:48) ( λ ) V f (cid:20) τ (cid:48) (1 − K ) − q f κ K (cid:21) + q f (cid:115) f κq τ (cid:48) ∂ λ V f (cid:27) = 0 , for f ( A ) f (cid:48)(cid:48) + (4 − q (cid:48) q ) f (cid:48) = q ˆ n w e − A (cid:115) f κτ (cid:48) /q ˆ n + ( e A wV f ) = − q ˆ ne − A Φ (cid:48) (2.33)and for τ ( A )(1 + K ) τ (cid:48)(cid:48) − (cid:18) q f κ + τ (cid:48) (cid:19) ∂ ln V f ∂τ + f κ q (cid:20) d ln f κdA + 2 λ (cid:48) ∂ ln V f ∂λ + K (cid:18) d ln f /κdA (cid:19)(cid:21) τ (cid:48) + (cid:20) − q (cid:48) q + d ln f κdA + λ (cid:48) ∂ ln V f ∂λ + K (cid:18) − q (cid:48) q + f (cid:48) f (cid:19)(cid:21) τ (cid:48) = 0 . (2.34)Here K = K ( A ) = ˆ n e A V f w . (2.35)This has the formally notable consequence that the A -equations are not autonomous;there is explicit A dependence. The consequence of this will become explicit when weconsider the scaling properties of the solutions in the following section; see Eq. (2.42).The equation (2.22) for f can be integrated once:˙ f ( z ) = 1 b ( z ) (cid:20) C + (cid:90) z du ˆ n b w (cid:115) fκb ˙ τ ˆ n + b w V f (cid:21) = 1 b [ C + ˆ n ( µ − Φ( z ))] , (2.36)– 11 –sing (2.16). Then f ( z ) is obtained by one more integration, with integration constantsdetermined by f (0) = 1, f ( z h ) = 0. Actually we are most interested in the chargedblack hole temperature, for which one obtains4 πT = − ˙ f ( z h ) = 1 − ˆ n (cid:82) z h du Φ( u ) b ( u ) b ( z h ) (cid:82) z h dub ( u ) . (2.37) Numerical solutions have to be transformed to the required standard form by usingscaling properties of the equations. A thorough discussion is given in Appendix B, andwe summarize the main points in the following. The quantities which are not mentionedwill remain unchanged and all bulk fields are taken to be either functions of z or A .For the z equations (2.20)–(2.23) one performs the following scalings: • The boundary value of f ( z ) must be set to 1 so that the boundary metric is pureAdS, f (0) = 1. This is achieved by scaling f → ff , f ≡ f (0) . (2.38)In order to keep b /f and K ( z ) in (2.17) invariant, this requires that further b → b √ f , ˆ n → ˆ nf / , ˙Φ → f ˙Φ . (2.39)Note that also the integration constant ˆ n is scaled. • The unit of energy can be changed by z → Λ z , together with b → b Λ , ˆ n → ˆ n Λ , ˙Φ → ˙Φ , (2.40)which leave the equations of motion invariant.For the A equations (2.31)-(2.34) the corresponding scalings are: • Scaling of f to f = f ( ∞ ) = 1 requires that q /f be constant, so that f → ff , q → q √ f . (2.41)Note that the density factor K ( A ) in (2.35) is not affected by this scaling. • The scaling corresponding to z → Λ z is A → A − ln Λ , b = e A → b . (2.42)The invariance of the density factor K ( A ) and Eq. (2.33) then demand thatˆ n → ˆ n Λ , Φ (cid:48) →
1Λ Φ (cid:48) . (2.43)– 12 – Constant Scalar Solutions and IR Stability
To gain intuition on what to expect at zero temperature and finite chemical potentialwe now consider some special solutions of the equations of motion derived in Sec. 2.We need to determine the fixed point solutions with translational symmetry since flowsbetween different such solutions categorize the various RG flows of the boundary theory.In general the fixed point solutions with translational symmetry are AdS p solutionseither with fixed scalars or hyperscaling violating solutions when the scalars run off toinfinity, [37, 40].We have not found hyperscaling violating asymptotics in this theory. The otherremaining scaling solutions must then have constant scalars. These solutions will bethe non-linear generalization of AdS Reissner-Nordstr¨om black hole (the so-called DBIblack hole), and solutions with scaling AdS regions in the IR, at extremality.To search for these, we turn to the equations of motion and make the followingreplacements: λ ( z ) → λ , τ ( z ) → τ and V f , V g , κ, w → V f , V g , κ , w (3.1)where a zero sub- or superscript indicates the constant value of the appropriate quantityin the fixed point and V f ≡ V f ( λ, τ ) (cid:12)(cid:12)(cid:12) λ = λ , τ = τ , ∂ λ V ≡ ∂ λ V f ( λ, τ ) (cid:12)(cid:12)(cid:12) λ = λ , τ = τ , etc . (3.2)The two classes of solutions are distinguished by whether the scale factor A is constantor not. If it is constant we obtain AdS type solutions while if it is non-trivial we obtainAdS type solutions. and the DBI Black-Hole Solution For constant scalars many of the equations become quite simple, and often can bedecoupled. For example, the equation governing the warp factor, A = log b is just A (cid:48)(cid:48) ( z ) − A (cid:48) ( z ) = 0 , (3.3)which has two independent solutions, A ( z ) = − log z or A constant. The first matchesthe AdS result in these coordinates. This is the solution one anticipates as a UV fixedpoint in the dual theory. It will turn out to be the charged DBI black hole, whichbecomes the AdS Reissner-Nordstr¨om solution in the limit of small gauge coupling.We can systematically insert this solution into the remaining equations of motion.– 13 –he Maxwell equation in this limit readsΦ (cid:48) ( z ) ≡ E ( z ) = − ˆ n zV f w (cid:113) ˆ n z V f w (3.4)and from this we obtain the behavior of the blackening function f in the uniform scalarbackground. This function is described by the equation of motion f (cid:48)(cid:48) ( z ) − z f (cid:48) ( z ) = ˆ n V f w (cid:113) ˆ n z V f w z . (3.5)The solutions of the correponding homogeneous equation give the standard blackeningfor the AdS black hole in five dimensions. The general solution of the inhomogeneousequation therefore takes the form f ( z ) = c − z z + Q ( z ) , (3.6)familiar for charged black holes. Here c and z are integration constants. The function Q , which carries the information about the electric source for the black hole, can becomputed by integrating (3.5) twice. It turns out to be (see also Eq. (F.7)) Q ( z ) = − V f (cid:115) n z V f w + 18 ˆ n V f w z F ( 13 , ,
43 ; − ˆ n z V f w ) , (3.7)where F is the hypergeometric function. For consistency, the blackening functionmust be compatible with the constraint equation (2.24), given by f (cid:48) ( z ) − z f ( z ) = 13 z (cid:32) V f (cid:115) n z V f w − V g (cid:33) . (3.8)Note that this equation effectively governs the constant term in (3.6), or equivalentlythe near boundary value of the blackening function. To leading order in ˆ n the solutionconsistent with the above constraint is f ( z ) = 112 (cid:0) V g − V f (cid:1) − z z + O (ˆ n ) (3.9)which is exactly the form one would anticipate in AdS with charged branes. Notice that when computing the full RG flow we have chosen to normalize f to one in the UV.We have the freedom to do this if a constant term is also included in the UV solution for A . – 14 –he remaining equations of motion, those for the dilaton and tachyon, containalgebraic constraints for various parameters of the theory. Specifically, the dilatonequation implies λ = 0 or ∂ λ V g = ∂ λ V f = ∂ λ w = 0 (3.10)while the tachyon equation needs either ∂ τ V f = ∂ τ w = 0 or V f κ = ∞ (3.11)in order to be satisfied. These constraints have a simple interpretation. The set ofequations ∂ λ V g = ∂ λ V f = ∂ τ V f = 0 (3.12)are simply the requirement that all the potentials are extremized at the appropriatevalue of ( λ , τ ). Evidently, the same must be true for the gauge kinetic function w ( λ, τ ).For the V-QCD potentials of interest, specifically those from Section 2, it turns outthat the extremization condition in 3.10 can never be realized and the only possibilityis the vanishing dilaton, λ = 0. The gauge kinetic function w and the flavor potential V f are of the general form of Eq. (2.4) and Eq. (2.6), respectively: V f = x f v f ( λ ) e − a ( λ ) τ and w = w ( λ ) (3.13)so ∂ τ w = 0 and the tachyon constraint reduces to0 = ∂ τ V f ∼ − x f τ a ( λ ) v f ( λ ) e − a ( λ ) τ . (3.14)Therefore, the flavor potential is extremized in the τ direction for either τ = 0 or τ = ∞ . Moreover, it can be explicitly checked that when the dilaton is zero there isno location in the parameter space ( x f , τ ) for which the combination V f κ diverges.Accordingly, one finds that in this V-QCD setup DBI black hole solutions exist at all x f so long as λ = 0 and τ = 0 or ∞ . Solution
There exists another simple solution to the constant scalar warp factor equation ofmotion (3.3). This is the constant solution A = A . In this case, the Maxwell equationis satisfied by a constant electric field of the form E = − ˆ n e − A w V f (cid:113) e − A ˆ n V f w (3.15)– 15 –iving a potential Φ( r ) = µ + E r (3.16)which is the correct form for a gauge field in AdS . The equation for the blackeningfunction is f (cid:48)(cid:48) ( r ) = e − A w V f ˆ n (cid:113) e − A ˆ n V f w (3.17)and has the general solution f ( r ) = C + C r + 12 e − A w V f ˆ n (cid:113) e − A ˆ n V f w r (3.18)The AdS solution is simply the one in which C = C = 0, and we identify the AdS radius, L , as L = 2 e A w V f (cid:113) e − A ˆ n V f w ˆ n (3.19)All the rest of the equations simply give constraints that determine when this solutioncan be realized. The “zero energy” constraint says that0 = V g − V f (cid:115) n ( e A V f w ) (3.20)while the dilaton equation of motion requires0 = ∂ λ V g − (cid:113) ˆ n ( e A V f w ) (cid:32) ∂ λ V f − V f w ˆ n ( e A V f w ) ∂ λ w (cid:33) (3.21)and the tachyon equation forces0 = ∂ τ V f − V f w ˆ n ( e A V f w ) ∂ τ w . (3.22)In the following section we will investigate these constraints in more detail, to determinewhether or not they can be realized in V-QCD models of interest. Note that we have anticipated the fact that the bulk geometry will be different from the DBI blackhole by employing a new radial variable r . For the solution of this section the IR limit is r → r → ∞ . – 16 – .3 A closer look at the AdS solution We can summarize the AdS requirements succinctly by recalling the definition of theeffective potential, Eq. (2.25), in the language of this section: V eff ( λ, τ ) = V g ( λ ) − V f ( λ, τ ) (cid:115) n e A V f ( λ, τ ) w ( λ ) (3.23)in which case the AdS constraints are simply V = ∂ λ V = ∂ τ V = 0 . (3.24)The zero energy constraint V = 0 shows that the volume form on the R factor is justVol R = e A = | ˆ n | w (cid:113) V g − V f (3.25)so one can think of this condition as an expression describing the size of the R , asdetermined by the values of the potentials at the fixed point. For the class of potentialsof immediate interest, this relationship fixes the volume of R in terms of ( λ , τ , x f , ˆ n ).Solving the zero energy constraint for ˆ n allows one to rewrite the extremizationconditions like 0 = ∂ λ log (cid:104) w (cid:0) V g − V f (cid:1) (cid:105) (3.26)0 = ∂ τ log (cid:104) w (cid:0) V g − V f (cid:1) (cid:105) (3.27)Note that these expressions depend only on x f , λ , and τ , and that the notation asksone to differentiate the potentials first, then evaluate the result at the constant scalarsolution.Finding simultaneous solutions to these equations provides the parameter space ona two-parameter plane in which the AdS solution can be realized. For the class ofpotentials used in V-QCD (3.13), this constraint is again trivially satisfied for τ = 0or τ = ∞ . For vanishing τ , it is easy to find solutions to the constraint numericallyfor the V-QCD potentials in Section 2. They appear in figure 2. Interestingly, there isa region at low x f where there are two solutions for constant (positive) dilaton. Thisbehavior may be an artifact of the parametrization of the potential w ( λ ). The secondfixed point is not expected, but we also find that it plays no role in the phase diagram.In the case of the divergent tachyon, τ = ∞ it is clear that V f = 0. One can carryout the same analysis as in the τ = 0 case to search for allowed AdS solutions inV-QCD, carefully navigating the somewhat subtle limits implied by this solution. For– 17 – (cid:230) n (cid:96)Λ (cid:120) Λ Figure 2 . Allowed (ˆ n, λ ) (left) and ( x, λ ) (right) values for the AdS solution withvanishing tachyon. At left, the black dots mark the location of x f = 1 along each branch,and correspond precisely to the AdS solutions found numerically and shown in figure 7. Theblack dashed line marks the Banks-Zaks limit at x f = 11 /
2. From the right plot we find thatwhen x f (cid:38) .
865 the constant dilaton solution becomes negative and is thus excluded as afixed point candidate. finite ˆ n but vanishing V f one finds that a divergent tachyon implies an electric field(3.15) and AdS radius of the form E = − ˆ n | ˆ n | e A w and L = 2 w | ˆ n | e A (3.28)The extremization condition (3.26) becomes0 = ∂ λ log( w V g ) (3.29)and the numerical results for the potentials in Section 2 are shown in figure 3. As– 18 – (cid:120) Λ (cid:120) Λ Figure 3 . Allowed ( x, λ ) values for the AdS solution with divergent tachyon. The leftplot shows the small branch of solutions, which cease to exist for x f (cid:38) . x f = 11 / before there are two branches of solutions—the smaller of which terminates at somefinite value of x f within the Banks-Zaks limit at x f = 11 / Region
The AdS solutions can be a priori endpoints or starting points of RG flows. Todetermine exactly what happens we must do a scaling analysis of the perturbationsaround them.We perturb the background AdS metric liked s = − D ( r )d t + B ( r )d r + C ( r )d (cid:126)x (3.30)where D ( r ) = r L (cid:16) D r d (cid:17) (3.31) B ( r ) = L r (cid:16) B r b (cid:17) (3.32) C ( r ) = C + C r c (3.33)In this background, the IR is approached as r → r → ∞ . Here L is the AdS radius as given by (3.19), C controls the volume of the R factor, andthe other constants parametrize the fluctuations in the obvious way. Without loss ofgenerality, we set C = 1 in what follows. All fluctuation amplitudes are taken to besmall. – 19 –he background fields are perturbed as well, λ ( r ) = λ + λ r a (3.34) τ ( r ) = τ + τ r t (3.35)Φ( r ) = µ + r (cid:16) E + Φ r f (cid:17) (3.36)The program is to insert these perturbation Ans¨atze into the equations of motion,linearize the equations about the fluctuations, and subsequently determine the scalingexponents and the fluctuation amplitudes that describe a given perturbation.Operationally, one first sets all the fluctuations above the background proportionalto the same power, which is to say α = d = b = c = a = t = f (3.37)The linearized fluctuation equations then reduce to a coupled set of homogeneous linearequations in the amplitudes of the fluctuations F i = { D , B , C , λ , τ , Φ } . Impor-tantly, the radial Anstze under investigation leaves a residual gauge freedom relatedto reparametrizations of r . Practically, this means that fixing B constitutes a gaugechoice, and the linear system consists of 5 independent equations. Requiring that thesystem have a non-trivial solution is equivalent to requiring that the determinant ofthe matrix of coefficients, M vanish for all r .In this case, one finds that the determinant is of the formdet M = α ( α − α + 1) ( α + 2) g ( α, λ , E ) (3.38)which vanishes for α ∗ = { , − , − , } and for the α = α ∗ such that g ( α ∗ , λ , E ) = 0.The former correspond to “universal” modes, while the latter are “non-universal” inthe sense that they depend on the details of the various potentials. Of the universalmodes, we find that there are two types of IR relevant ( α <
0) modes in the fluctuationspectrum, with exponents α ∗ = {− , − } . That they correspond to relevant operatorsin the IR is clear from the fact that when α ∗ < r → n can be easily related to the boundary valueof the electric field ( E in this section) via (3.15). The effective potential also turns outto govern the properties of two of the four non-universal exponents, α λ ± = − (cid:34) ± (cid:114) − λ L ∂ λ V (cid:35) (3.39)– 20 – (cid:45) (cid:120) Α (cid:45)Λ (cid:45) (cid:45) (cid:45) (cid:45) (cid:120) Α (cid:43)Λ Figure 4 . The numerical values of non-universal exponents α λ ± from (3.39), for the solutionswith vanishing tachyon. Relevant operators have negative exponents in this analysis. Thelarge λ branch of solutions is colored purple. The BF-like bound mentioned in the text isnever exceeded. The red dashed line indicates x f ≈ .
865 , beyond which the small branchof constant dilaton AdS solutions vanishes. The domain of x f terminates at the Banks-Zakslimit x f = 11 / while the other two are α τ ± = − ± (cid:115) E w (cid:112) − E w κ L ∂ τ V f (3.40)The superscripts signify the fact that these modes correspond to perturbations of theappropriate scalars as we will see below.These exponents have a few noteworthy features. First, all of the exponents —universal or not — can be pairwise summed to give α + + α − = −
1, which is the correctstructure for modes in AdS , in these coordinates. Moreover, we see from (3.39) thatthere is a BF-like bound signaling the onset of an instability when ∂ λ V >
23 1 L λ . Forthe V-QCD potentials employed for numerical studies, these non-universal exponentsare plotted in figures 4, 5 and 6 as functions of x f for both branches of the AdS fixedpoint. Evidently, while the BF-like bound is never exceeded in the fluctuations corre-sponding to α λ , the fluctuation characterized by α τ realizes an analogous instabilityaround x f ∼ . τ case. When the tachyon is divergent, the equa-tions of motion require E w = 1 and thus α τ ± saturates to {− , } . It will turn outthat the fluctuations described by α τ are appropriately named, as they correspond tofluctuations of the tachyon alone.The full description of the perturbation is given by the exponent α ∗ , which containsinformation about the dimension of the dual IR operator, and the amplitudes of the– 21 – (cid:45) (cid:45) (cid:120) Α (cid:45)Λ (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:120) Α (cid:43)Λ Figure 5 . The numerical values of non-universal exponents α λ ± from (3.39), for the solutionswith divergent tachyon. Again, the large λ branch of solutions is colored purple and the BF-like bound is never exceeded. The red dashed line indicates x f ≈ .
685 , beyond which thesmall branch of constant dilaton AdS solutions vanishes. (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:120) Α (cid:45)Τ (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) (cid:120) Α (cid:43)Τ Figure 6 . The numerical values of non-universal exponents α τ ± from (3.40), for the solutionswith vanishing tachyon. Relevant operators have negative exponents in this analysis. Thelarge λ branch of solutions is colored purple. Note that for x f larger than approximately2.4 the dual operator in the large branch fixed point has complex dimension, signaling aninstability (marked by green dashed line). The red dashed line indicates x f for these potentialsbeyond which the small branch has λ <
0. Again, the domain of x f terminates at the Banks-Zaks limit x f = 11 / various modes that are activated by this fluctuation. The following cases are pertinentfor the two conjugate solutions: • If the operator is UV relevant then both perturbations vanish in the UV boundary. • If the operator is IR relevant then both perturbations blow-up in the IR regime. • If the operator is UV irrelevant then one perturbation vanishes and one blows upin the UV boundary. – 22 –
If the operator is IR irrelevant then one perturbation vanishes and one blows upin the IR regime.The amplitudes are easily obtained by solving the linear system provided by a given α ∗ , and in general depend on one undetermined (but non-vanishing) amplitude and achoice of radial gauge which can be fixed via B . The results are listed in Appendix A.We conclude this section by assessing the RG stability of AdS solutions. The onethat appears at small values of λ , denoted by a blue line in figure 2 has dilaton andtachyon perturbations that render it IR unstable. This explains the fact that it playsno role in the phase diagram we describe in this work.The other AdS solution that corresponds to the purple line in figure 2 has dilatonand other perturbations that are IR irrelevant but the tachyon perturbation is IRrelevant in the non-tachyonic black-holes. This is as expected as we need to tune m q = 0 to reach this solution in the IR. Once we turn on m q (cid:54) = 0 we will avoid itand end up in the tachyonic black hole. On the other hand in the tachyonic case, thedilaton perturbation is IR irrelevant and the tachyon one is marginal. However it doesnot correspond to an extra parameter in the theory as τ = ∞ is a singular point infield space. The equations of motion admit two types of solutions at finite temperature and chemicalpotential, which we call black hole and thermal gas solutions. The thermal gas solutionshave no horizon in the IR. In this case the temperature is identified as the inverse ofthe length of the compactified time coordinate, while Φ = const . = µ . The blackeningfactor is trivial, f ≡
1, and the z -dependence of the other fields is exactly the same asfor the solutions at T = 0 = µ , which were constructed in [33]. When 0 < x f < x c ,the dominant vacuum was found to have a nonzero tachyon field (and therefore brokenchiral symmetry). The thermodynamics of the corresponding thermal gas solution istrivial: the pressure is independent of T and µ and will be normalized to zero here.Likewise, the condensate, which signals chiral symmetry breaking, will be nonzero but T independent.The nontrivial task on which we concentrate in this article is the construction ofthe black hole solutions. The equations we have to solve numerically are the Einstein’sequations (2.31) and (2.33), the equations of motion for λ , equation (2.32), and theequation of motion for τ , equation (2.34). Their solution for ˆ n = 0 has been discussedin detail in [34]. The numerical solving with given initial conditions as such is very– 23 –imple using NDSolve of Mathematica. The main issue is the correct initialization andsubsequent processing of the solutions via the scalings described in section 2.4.An important general feature is that there are two types of black hole solutions: • The solutions with τ = 0 which describe the hot and dense matter in a chirallysymmetric phase; these are expected to dominate the free energy at large T or µ . • The solutions with τ ( A ) (cid:54) = 0. These will describe a chirally broken phase, ex-pected to dominate at small T or µ . These solutions are parametrized by thevalue of the quark mass m q = lim A →∞ L − τ ( A ) e A ( A − ln(Λ L UV )) γ /b . (4.1)Since we are interested in solutions with exact chiral symmetry, we need to restrict to m q = 0. This is a technically very demanding task (see Appendix C) and necessitatesgoing to very small values of z ≈ e − A , up to A ∼ hundreds. This is one of the reasonsfor using A as a coordinate. The details of the numerical solution and the associatedscaling properties are discussed in detail in Appendix B.In the numerical computations we choose the unit of number density so that L A =1. In section F we shall actually fit that L A ≈ . For thermodynamics one needs solutions with a black hole. To generate them numer-ically, one has to start the integration at the horizon, which we place at A = A h suchthat f ( A h ) = 0. Because of the singularities due to the 1 /f terms in Eqs. (2.31)-(2.34)one cannot start the integration precisely at the horizon. Instead, one first writes thevalues of the fields at a small distance (cid:15) from the horizon by expanding in (cid:15) as q = q h + (cid:15)q (cid:48) h + O ( (cid:15) ) , (4.2) λ = λ h + (cid:15)λ (cid:48) h + 12 (cid:15) λ (cid:48)(cid:48) h + O ( (cid:15) ) , (4.3) f = (cid:15)f (cid:48) h + (cid:15) f (cid:48)(cid:48) h + O ( (cid:15) ) , (4.4) τ = τ h + (cid:15)τ (cid:48) h + (cid:15) τ (cid:48)(cid:48) h + O ( (cid:15) ) , (4.5)which are then inserted to the equations of motion. Here and in the following thesubscript h denotes quantities evaluated at the horizon. Then one expands in (cid:15) anddemands that the divergences and the constant term vanish. Note that the input hereis that in (4.4) f h = f ( A h ) = 0.Out of the leading terms in (4.2)-(4.5) one can choose f (cid:48) h = +1 as the magnitude of f ( A ) will anyway be fixed by the scaling (2.41) to the boundary value f ( A → ∞ ) = 1.– 24 –he dilaton value at the horizon λ h will remain as a parameter, closely associated withtemperature. The second parameter, closely related to the chemical potential, is ˆ n .However, in the numerics it turns out to be more practical to use instead˜ n = e − A h ˆ n = ˆ nb h (4.6)which is invariant in the scaling of (2.42) and (2.43). The tachyon value at the horizonwill be fixed by the quark mass, τ h = τ h ( λ h , ˜ n ; m q ). Including the terms up to O ( (cid:15) )for q and up to O ( (cid:15) ) for the other fields in (4.2)-(4.5) is sufficient to ensure that thevalues of these parameters in the resulting numerical solution match with their inputvalues to a high precision.The remaining first-order derivative terms will be fixed by demanding that the 1 /A (i.e. 1 /(cid:15) ) singularities cancel. Canceling the divergent 1 /A term of (2.31) gives q h = − (cid:112) f (cid:48) h (cid:113) V g − V f √ K h , K h = ˜ n w V f . (4.7)with the understanding that the potentials V g , V f , and w are evaluated at the horizon.Canceling the divergent 1 /A term of the λ equation (2.32) gives λ (cid:48) h = − λ h q h f (cid:48) h √ K h (cid:18)(cid:112) K h ∂ λ V g − ∂ λ V f + K h V f κ (cid:48) h κ h (cid:19) = − λ h q h f (cid:48) h ∂ λ V eff ( λ h , τ h , ˜ n ) (4.8)and canceling the 1 /A term of the τ equation gives τ (cid:48) h = q h ∂ τ ln V f f (cid:48) h κ h (1 + K h ) . (4.9)In (4.7) and (4.8) we again have the important quantity, V eff = V g ( λ ) − V f ( λ, τ ) (cid:115) n b w V f = V g ( λ ) − V f ( λ, τ ) (cid:115) n b h b w V f , (4.10)evaluated at the horizon.This leaves us with the four quantities f (cid:48)(cid:48) h , q (cid:48) h , λ (cid:48)(cid:48) h , τ (cid:48)(cid:48) h to be determined by requiringthat the constant terms of the four equations vanish. The constant term of equation(2.33) gives a simple relation between f (cid:48)(cid:48) h , q (cid:48) h : f (cid:48)(cid:48) h + f (cid:48) h (cid:18) − q (cid:48) h q h (cid:19) − K h √ K h q h V f = 0 . (4.11)– 25 –he remaining expressions are too complicated to be reproduced here but can be foundin [39]. From the algebraic derivation of the initial conditions to the numerical integra-tion of the system of differential equations (2.31)-(2.34), we treat the whole problem inMathematica. It is thus easy to produce some numerical solutions for the functions q ( A ), λ ( A ), f ( A ),and τ ( A ) with Mathematica, given λ h and ˜ n , but an essential and nontrivial part of thenumerical work is to transform the solutions to a standard form satisfying in z coordi-nates f (0) = 1 and that the scale of the UV expansions equals one (see Appendix B).In A coordinates these conditions become lim A →∞ f ( A ) = 1lim A →∞ (cid:18) b λ ( A ) + b b ln( b λ ( A )) − A (cid:19) = − ln L UV . (4.12)The former is implemented by scaling f as in (2.41), the latter by scaling A as in (2.42).To achieve this one determines the scaling factor Λ( λ h , ˜ n ) so that the asymptotic limit(4.12) holds. We start from a numerical solution having A h = 0, then according to(2.42) the value of b at the horizon in the scaled solutions is simply given in termsof the scaling factor by b h = exp( A h ) / Λ = 1 / Λ. From the standard configurations soobtained one then computes the temperature as the black hole temperature and thechemical potential using (2.16) and (2.19), otherwise the configurations as such are notof interest for this calculation. The procedure is described in detail in Appendix B.Summarising, from the numerical integration of equations of motion, for given( λ h , ˜ n ), one obtains the following quantities: b h ( λ h , ˜ n ) , T ( λ h , ˜ n ) , µ ( λ h , ˜ n ) . (4.13)From these we obtain the entropy density using the basic formula s ( λ h , ˜ n ) = A G = b h G . (4.14)To obtain the 4d physical quark number density note first that, when deriving theΦ equation of motion from the fermionic part of the action, one has, for solutions ofequations of motion, δS f = 116 πG VT (cid:90) z h (cid:15) dz ddz (cid:18) ∂L f ∂ ˙Φ δ Φ (cid:19) . (4.15)– 26 –t z h one has to keep the value Φ( z h ) fixed to zero so that δ Φ( z h ) = 0 and the upperlimit does not contribute. Since S = − Ω /T and δ Φ = dµ the fermionic contributiongiven by the above integral is the n dµ term in the free energy, and therefore the correctnormalization of n is n = ˆ n πG = ˜ n b h πG = ˜ n π b h G = s ˜ n π , (4.16)where we used the definition (4.6). This expression also gives a physical interpretationof the parameter ˜ n of the integration of the equations of motion:˜ n = 4 π ns . (4.17)Next we discuss what values of ( λ h , ˜ n ) are possible and how the pressure is inte-grated from dp = s dT + n dµ . To compute the pressure we have to integrate over T and µ and these one-dimensional integrals are most simply carried out by convertingthem into integrals over λ h at fixed ˜ n or vice versa, see Section 5.1. λ h , ˜ n plane For ˜ n = 0 one found (see, e.g., [34], Fig. 7) that chirally symmetric solutions, i.e. theones with zero tachyon, existed only for 0 < λ h < λ ∗ , with λ ∗ given by the extremumof the effective potential in (4.10), and chirally broken solutions with nonzero tachyonexisted only for λ h > λ end with 0 < λ end < λ ∗ . This is in harmony with the expectationthat large T and chiral symmetry are associated with small coupling, small λ h , andstrong coupling leads to chirality breaking. The introduction of ˜ n extends this patternin an interesting and subtle way, exhibited in Fig. 7.For small ˜ n the above pattern remains unchanged, only the curves λ end (˜ n ) and λ ∗ (˜ n ) slowly decrease. Also thermodynamically the situation is only slightly modified:as we shall soon see along λ end the symmetric and broken phase pressures are equaland there is a continuous chiral phase transition. This pattern continues all the way tothe point labeled AdS in Fig. 7, where ˜ n = ˜ n max ≈ . λ end from the small ˜ n -side, temperature goes to zero. When ˜ n > ˜ n max , sincethere is no solution with τ h = 0, the chiral symmetry breaking solution at the limit λ end cannot, and does not, have τ h →
0, and cannot therefore be a part of the secondorder transition line. The temperature, when approaching that line, does decrease tovery small values, but unfortunately the numerics is not stable enough to be confidentthat it indeed converges to zero. The hypothesis is, however, that it indeed does, andthat the ˜ n > ˜ n max section of the λ end curve consists of T = 0 chiral symmetry breakingsolutions of various µ . – 27 – (cid:42) , T = 0 Λ end , Λ Χ b V ' eff = 0AdS AdS V eff = 0 n (cid:142) Λ h Figure 7 . The physical region on the λ h , ˜ n plane for chirally symmetric (red region)and chirally broken (blue region, unbounded above) solutions. Chirally symmetric regionis bounded from above by the curve λ ∗ (˜ n ) along which T = 0 up to the point AdS at˜ n = 12 . , λ h = 1 . V eff = 0 up to the sec-ond AdS point at ˜ n = 10 . , λ h = 0 . n = 10 . , λ h = 0.Tachyonic chiral symmetry breaking solutions exist only above the blue curve λ end (˜ n ) ≡ λ χb .The dashed lines are V eff = 0 and V (cid:48) eff ( λ h ) = 0 at τ = 0 (see (4.10)). The computation of the curves in Fig. 7 is mostly numerical, but parts of theboundary of the symmetric phase can be found analytically. Note first that (4.7)implies that V eff ( λ h , τ h ) = V g ( λ h ) − V f ( λ h , τ h ) (cid:112) K ( A h ) = V g ( λ h ) − (cid:115) V f ( λ h , τ h ) + ˜ n L A κ h > , (4.18)where we used (4.6) and inserted w from (2.4). For the symmetric phase τ h = 0 onecan solve from here the upper boundary for values of ˜ n :˜ n ≤ ˜ n max = L A κ h (cid:113) V g ( λ h ) − V f ( λ h , . (4.19)This with L A = 1 is the curve V eff = 0 in Fig. 7. Further, due to the interpretation β ( λ ) = λ (cid:48) ( A ) one usually expects that λ ( A ) monotonically decreases from its value λ h = λ ( A h ) towards λ ( A = ∞ ) = 0, and in particular that λ (cid:48) h <
0. From (4.8) this– 28 –ould imply ∂ λ V eff ( λ h , τ h ( λ h , m q ) , ˜ n ) ≥ . (4.20)However, deep in the IR the interpretation of λ (cid:48) ( A ) as a negative beta function neednot be valid and solutions with signs opposite to those in (4.20) are also possible. Thisis confirmed by numerical computation and the real boundary is given by finding where T = 0 or where the scale factor Λ( λ h , ˜ n ) diverges. Requiring that both V eff and V (cid:48) eff vanish has two solutions marked AdS , since actually the geometry at these points isasymptotically AdS × R in the IR. The lower AdS point disappears at larger x f .The boundary of the broken phase marked λ end in Fig. 7 is discussed in some detailin Appendix D. It is a lower limit for possible values of λ h . For ˜ n there is an upperlimit, but there is no upper limit for λ h .The significance of various parts of the physical region is also described by plottingcurves of constant T and µ as in Fig. 8. Actually we show there the result only forthe discrete values of ˜ n used in the pressure integration. Particularly interesting isthe behavior of the µ = constant curves. Extrapolating them one sees that clearlyasymptotically µ = 0 in the upper part of the T = 0 curve. States here have T = µ = 0and thus represent vacuum. In the vertical part of the T = 0 curve correspondingly µ = ∞ . This is also some special state. All the µ = constant curves end at the AdS point, where thus all the exactly T = 0, µ finite symmetric phase thermodynamicsresides.A third important quantity is the dimensionless scale factor Λ( λ h , ˜ n ). It varies alot as a function of λ h . The main part of the variation is contained in Λ( λ h ,
0) whichat the boundaries of the phase space, λ h → λ h → λ ∗ (0), can be fitted byΛ( λ h ,
0) = 0 . e − / ( b λ h ) ( b λ h ) b /b (1 + 2 . λ h + · · · ) = z h L UV , (as λ h → . (cid:112) λ ∗ (0) − λ h + 2 + · · · (as λ h → λ ∗ (0)) . (4.21)Fig. 9 shows curves of constant Λ( λ h , ˜ n ) / Λ( λ h , b h = 1 / Λ, s ∼ b h , n = s ˜ n/ (4 π ),also curves of constant entropy and number density are contained in this figure. Onesees that the dependence on ˜ n is surprisingly weak except at the T = 0 boundaries.From the figure one can extrapolate that • On the upper T = 0 boundary from ˜ n = 0 to the AdS point: T = µ = s = n = 0,Λ = ∞ . So this really is the vacuum. All the T = 0, finite µ chirally symmetricmatter is exactly at the AdS point • On the vertical T = 0 boundary between the two AdS points T = Λ = 0, µ = ∞ , s = ∞ , n = ∞ , n/s = ˜ n/ (4 π ), 10 . < ˜ n < . igure 8 . Constant values of T and µ on the ˜ n, λ h plane for tachyonless solutions for thediscrete values of ˜ n used in the pressure integration, other boundary curves as in Fig. 7. Small λ h corresponds to large T as expected. The special role played by the red AdS point is seen:above it along the boundary curve µ = 0, below it µ = ∞ . Thus effectively at the AdS pointall positive values of µ are obtained. A straightforward way to generate the data necessary for solving the thermodynamicswould be to compute black hole solutions in both the symmetric and non-symmetricbranch of the solutions on a sufficiently dense lattice in the physical region of the(˜ n, λ h ) -plane. In order to carry out the pressure integrals without accumulating largecumulative errors, a reasonably accurate continuum interpolation of the observables isneeded. Since the observables as a function of (˜ n, λ h ) are mostly smooth but have linesof zeroes and divergences, detailed in the following section, it would be necessary touse at least a somewhat sophisticated interpolation algorithm with an adaptive localgrid size in two dimensions, or alternatively a very large amount of computing powerfor the brute force approach of simply a very dense uniform lattice.However, we have been able to avoid constructing a full 2D interpolation of thesolutions by considering a grid of 1D interpolations, for which well-established adaptivealgorithms are readily available. The two primary interpolations are curves with ˜ n as aconstant and λ h as the variable, and those with λ h as a constant and ˜ n as the variable.We compute the interpolations for a number of values of ˜ n and a number of values of λ h .Figures 18 and 21 in appendix D show images of these curves of the ( µ, T ) -plane.We– 30 – igure 9 . Constant values of the scale factor Λ, normalised to its value at ˜ n = 0, on the ˜ n, λ h plane for tachyonless solutions for the discrete values of ˜ n used in the pressure integration,other boundary curves as in Fig. 7. Curves of constant s and n can be inferred from this (seetext). can then compute the pressure integrals along each of these, for both the symmetricand non-symmetric branches, fixing the constants as described in the next section.At least in the specific case handled in this paper, the constant parameter curvemethod has allowed us to extract all the thermodynamic features of interest withoutresorting to full 2D interpolation. However, in the case of a transition between tworegions of the same branch of solutions, such as happens at small x f for some ofthe potentials explored in [34], a complete 2D interpolation may become necessary toextract the phase transition line. According to the holographic dictionary, the pressure can in principle be computed byevaluating the on-shell action. However, this is numerically very challenging in this kindof model with corrections decaying only logarithmically near the boundary. Therefore,we use instead the usual thermodynamic formulas.We first review how the pressure is computed by integrating s ( T ) = p (cid:48) ( T ) for ˜ n = 0since this is how the constant of integration is fixed in [34] and will be fixed here, too.– 31 –ne has, see Fig. 10,4 G p b ( T ) = (cid:90) ∞ λ h ( T ) dλ h ( − T (cid:48) b ( λ h )) b hb ( λ h ) + p b ( ∞ ) , (5.1)4 G p s ( T ) = (cid:90) λ ∗ λ h ( T ) dλ h ( − T (cid:48) s ( λ h )) b hs ( λ h ) + p s ( λ ∗ ) , (5.2)where the subscripts b and s denotes quantities in chirally broken and symmetric phases,respectively. What matters for the phase structure is the difference of the integrationconstants p b ( ∞ ) and p s ( λ ∗ ). This is simply fixed by requiring that pressure be thesame for the two phases at λ h = λ end [34]. The outcome is plotted in Fig. 10. Atthis temperature there is a second order (both p and s ∼ p (cid:48) are continuous) chiralphase transition. The broken phase pressure vanishes at λ h = 3 .
19 at the temperature T h = T b (3 . ,
0) = 0 .
14. This is the deconfining transition. At higher λ h or smaller T the dominant phase is the thermal gas phase with vanishing thermal pressure.For quantitative correctly normalised results one will need both the energy unitΛ , which is implicit in formulas involving T and µ , and the constant 4 G . The formeris fitted by the value of the critical temperature T χ (0) = 0 . . Taking T χ = 0 . = 1 GeV . (5.3)For 4 G normalisation to the T Stefan-Boltzmann term at T → ∞ gives [34], seeEq. (F.14), 14 G = 445 π x f L N c = 445 π N c . (5.4)In general, we wish to obtain the pressure by integrating dp = sdT + ndµ . All thequantities on the RHS are numerically known as functions of λ h , ˜ n , see Appendix C.Note that we can write dp as dp = b h G dT + b h ˜ n πG dµ = b h G (cid:18) dT + ˜ n π dµ (cid:19) , (5.5)where all quantities are functions of λ h , ˜ n .The differential dp can now be integrated either over curves of constant ˜ n from λ b to λ t , 4 G p ˜ n ( λ t ) = (cid:90) λ t λ b dλ h b h ( λ h ; ˜ n ) (cid:20) T (cid:48) ( λ h ; ˜ n ) + ˜ n π µ (cid:48) ( λ h ; ˜ n ) (cid:21) , (5.6)or over curves of constant λ h from ˜ n b to ˜ n t ,4 G p λ h (˜ n t ) = (cid:90) ˜ n t ˜ n b d ˜ n b h (˜ n ; λ h ) (cid:20) T (cid:48) (˜ n ; λ h ) + ˜ n π µ (cid:48) (˜ n ; λ h ) (cid:21) . (5.7)– 32 – igure 10 . Left: The temperature as a function of λ h for the symmetric tachyonless andbroken tachyonic solutions at ˜ n = 0. Right: The pressures computed from (5.1) and (5.2)near the chiral transition region with relative normalisation fixed so that they are equal at λ end . The broken phase pressure vanishes outside the figure at λ h = 3 .
19 at the temperature T b (3 . ,
0) = 0 . To test the path dependence of the integral, one can choose an arbitrary rectanglewithin the physical region in Fig. 7 for either of the phases. This is mapped to a four-sided region on the grid in Fig. 21. One now integrates numerically around it usingEqs. (5.6) and (5.7) and checks whether the integral is zero. This is indeed what wefind to a great accuracy.This proof of path independence is a very impressive confirmation of the validityof our numerical computations. All the quantities included in (5.6) and (5.7) are theresults of lengthy numerical solutions of Einstein’s equations and it is striking to seethat when they are put together as above, the outcome is path independent to a verygood numerical precision.The pressures of the two phases p s ( λ h , ˜ n ) and p b ( λ h , ˜ n ) can now be computed byfixing the relative integration constant by demanding that p s ( λ end ,
0) = p b ( λ end ,
0) andintegrating to the point ( λ h , ˜ n ) along any convenient path. These can trivially beconverted to p s ( T, µ ) and p b ( T, µ ). Three-dimensional plots of pressure vs
T, µ arenumerically rather noisy and we shall focus on the main question: phase structure andphase transition lines.
We have discussed thoroughly the chirally symmetric and broken phases with pressures p s ( T, µ ) and p b ( T, µ ). Furthermore, as a model for the low T system we shall usethe thermal gas phase, for which the metric Ansatz is like that in (2.3) but with f ( z ) = 1 and T is introduced by compactifying the imaginary time region, otherwisethe equations of motion are as before. Note that from f = 1 and the equation of motion– 33 –2.22) it follows that one must have ˜ n = 0 so that also n = 0. Thus also ˙Φ = 0 so thatΦ = µ is constant. The property n = 0 for a low T chirally broken phase is consistentwith the fact that this model contains no baryons in the spectrum of singlet states.The pressure of the thermal gas phase is p low = 0. We identify this with the hadrongas phase of the field theory. This is well justified in the case of pure SU( N c ) gaugetheory, for which the pressure of the plasma phase is ∼ N c . In the case here (V-QCD)the degrees of freedom in the low temperature phase are the N f Goldstone bosons ofthe spontaneously broken chiral symmetry, while the high energy degrees of freedom arethe deconfined partons, 2 N c + 7 / N f N c . The ratio of the number of degrees of freedomat low and high temperature is then x f / (2 + 7 / x f ) which at x f = 1 is 0.18, and weexpect that taking p low = 0 provides still a useful guide towards the location of thedeconfinement phase boundary. However, as x f increases, the uncertainty associatedwith this approximation grows. At x f (cid:39) T χ ( µ ). At µ = 0 or ˜ n = 0 this took place at the point λ end in Fig. 10. As ˜ n is increasedtoward ˜ n max , the λ end -curve decreases monotonically and at λ end (˜ n max ) hits the AdS point. We find that the pressure along λ end is positive for the whole interval from ˜ n = 0to ˜ n max . Since the pressure decreases on the τ h (cid:54) = 0 -branch for increasing λ h , and isequal for the τ h = 0 and τ h (cid:54) = 0 -branches at λ end , the resulting chiral transition isbetween two stable phases. Along this transition line, the temperature also decreasesmonotonically, reaching T = 0 at the AdS point. It therefore divides the ( µ, T ) -planeinto a region near the origin, where chiral symmetry is broken, and an outside regionwith µ or T large, where chiral symmetry is restored.At ˜ n > ˜ n max , λ end (˜ n ) starts increasing again. The temperature along the curve isvery low and is consistent with being 0, but unfortunately the numerics is not quitestable enough to state this with certainty. The most likely option is that this is anunstable branch of chiral symmetry breaking T = 0 solutions, in analogy with the T = 0 -boundary of the chirally symmetric branch, as discussed in section 4.4.Below T χ ( µ ) the chirally broken phase is the stable one. Its pressure is positive butstarts decreasing when T is further lowered, λ h increased, just as happens in Fig. 10 at µ = 0. Computing the pressure of the broken phase for arbitrary values of ˜ n , one findsthat it vanishes for the values of λ h plotted in Fig. 22. The corresponding temperature T h ( µ ) is plotted in Fig. 11, see also Fig. 1. We interpret this as a first order deconfining– 34 – c H m L T h H m L m T H m L n T c H n L Figure 11 . Dependence of the chiral transition temperature on chemical potential or onquark number density. In the left panel p b > T χ and T h and vanishes along T h ( µ ).On it n jumps to zero. m= T ê T c H e- Lê T Figure 12 . The interaction measure (cid:15) − p = T s + µ n − p scaled by the ideal gas pressure(F.1) in the region T > T χ ( µ ) as a function of T /T χ (0) for µ = 0. The curve refers to thesymmetric phase and therefore starts at T /T χ ( µ ) = 1 (see also Fig. 1). transition between the chirally broken phase and the zero-pressure low T thermal gasphase. The transition temperature T h ( µ ) decreases monotonically with increasing µ ;our numerical accuracy does not permit to make definite statements about the limit T →
0. Note that this corresponds to very large values of λ h , see Fig. 22.One observes that at µ = 0 the temperatures T χ (0) and T h (0) = 0 . T χ (0) arevery close to each other. For reference, one may note that a very similar situationwith T h (0) = 0 . T χ (0) was observed in [41] in a completely different Schwinger-Dyson– 35 –quation model for QCD thermodynamics. There the conclusion was that the chiral anddeconfinement transitions probably coincide. These behaviors are most transparentlyunderstood on the basis of underlying exact and approximate symmetries and relatedorder parameters [42, 43].From the computed pressure, we can determine the interaction measure (cid:15) − p = T s + µ n − p = ( T ∂ T + µ∂ µ − p , which is shown close to the chiral transition regionin the symmetric phase in Fig. 12. Consider the curve for µ = 0, the structure ofwhich is described in Fig. 9 of [34]. The analogous curve for QCD is plotted, e.g.,in Fig. 3 of [44]. The V-QCD curve plotted in Fig. 12 starts by decreasing above T = T χ (0) but then changes direction and passes through a maximum at T ∼ T χ (0)with a QCD-like decay above that. This large T maximum can be interpreted [34] asa crossover transition. When x f is increased into the conformal region at x f > x c ≈ p/T which remains. It is now apparent thatincreasing µ does not change this overall pattern qualitatively. In particular, the large T decay is independent of the chemical potential. The chiral transition was above numerically observed to be of second order at µ = 0, forthe potentials used here. There are also potentials which lead to a 1st order transition,as concretely shown in [34]. It is commonly accepted that the chiral QCD transitionat N f ≥ N f = N c = 3, x f = 1 [46]. It is usefulto see how our gauge/gravity duality model, valid, in principle, for N c (cid:29) N f × N f matrix M ij ( x ) = (cid:104) q iL ¯ q jR (cid:105) , i, j = 1 , . . . , N f , x is the d = 3 dimensionalspatial coordinate. The potential term in the action is V ( M ) = m tr M † M + g (tr M † M ) + g tr M † M M † M. (5.8)To study the phase transition one should compute the effective potential of the theory.In the 1-loop approximation this was carried out, for m = 0, in [47]. Much informationcan already be obtained from the beta functions of the couplings in d = 4 − (cid:15) dimensions:if there is an infrared stable fixed point, zero of the beta function away from g = g = 0,the transition probably is of second order. If the couplings run to infinity, the transitionis of first order. In the computation of [47], the color and hence the value of N c ishidden in the color contraction in (cid:104) ¯ qq (cid:105) . Opening up these color interactions in the1-loop computation in full is an impossible task, but in the large N c limit a single tr is– 36 –lways one quark loop and thus suppressed by a factor 1 /N c . Thus we expect that inthe above effective potential g ∼ /N c and g ∼ /N c .According to [47] the β -functions of the two couplings in (5.8) (scaled by a factor π /
3) have a fixed point at N c g ∗ = 3 (cid:15)x f , g ∗ = 0 (5.9)with the eigenvalues (cid:15), − (cid:15) so that the fixed point is unstable, the flows are plotted in[47]. This is true also at large N c , N f , indicating a first order transition in this limit,too. However, one may argue that when N c = ∞ , the term with g in (5.8) shouldbe entirely neglected. Then only the β function for g remains and it has an infraredstable fixed point at N c g ∗ = 3 (cid:15) x f . (5.10)This indicates a 2nd order transition. The two arguments are compatible if the latentheat of the 1st order transition is ∼ /N c . Another way to say this is that as N c → ∞ and g /g →
0, the Hermitian model becomes equivalent to the O (2 N f ) model that isknown to have a second order phase transition.One should also remember that the (cid:15) expansion cannot give any definite answer.A good example of this is another standard model transition, the electroweak phasetransition. There the (cid:15) expansion method also leads to a first order transition [48]while a numerical computation leads to a first order transition for small Higgs masses, m H (cid:46)
75 GeV, while at larger Higgs masses there is only a cross over [49].
The quark number density as a function of µ is plotted in Fig. 13 for T = 0 .
1, as arepresentative value. As expected, n ( µ ) is continuous as the transition is of 2nd order.When plotted for T > T χ (0), the fixed- T curves contain just the symmetric phase withmonotonically increasing n ( µ ).It is common to characterize the µ dependence by the susceptibilities at µ = 0: χ ( T ) = ∂ p ( µ, T ) ∂µ | µ =0 , χ ( T ) = ∂ p ( µ, T ) ∂µ | µ =0 (5.11)In the ideal gas limit and for L A = 1, χ → x f N c T , χ → x f π N c . These (per N c ) arealso plotted in Fig. 13, now also including the normalisation term (see Appendix F).One is approaching the ideal gas limit but rather slowly.– 37 – .5 Polyakov line The basic difficulty in the study of thermodynamic deconfinement is that there is nosymmetry and thus no order parameter associated with deconfinement. For N f = 0the Polyakov line, trace of path ordered exponential of A over the periodicity range0 , /T in imaginary time, signals breaking of Z( N c ) symmetry and separates low andhigh T phases. Even though it is not an order parameter for finite N f , it is a gaugeinvariant measurable observable, which in lattice Monte Carlo studies varies togetherwith the chiral condensate (cid:104) ¯ qq (cid:105) .Constructing the gravity dual of the Polyakov line is very complicated, but we canmodel it in a simple way following [50]. The idea is to start from a duality determinationof a string tension in a thermal ensemble, interpret this as dF/dz = dF/dT × dT /dz ,compute from here dF/dT , integrate F ( T ) by choosing the integration constant byphysical arguments and finally plotting L = exp( − F ( T ) /T ). Here one can start fromthe determination of the spatial string tension σ s [51], determined from Wilson loopswith sides in spatial directions, σ s = 12 πα (cid:48) b h κ h = dFdz h = dTdz h dFdT . (5.12)Apart from the string tension all quantities here are known and the evaluation, using T = m H m ,T L Figure 13 . Left: A plot of the quark number density n ( µ ; T ) (without the normalisationfactor 4 / (45 π ) in (5.4)) for T = 0 .
1. At large µ the stable phase is always the symmetricphase (solid red) which can exist as a metastable phase (dotted red) even below the transition.Below the chiral transition the broken phase is stable (blue) and at the lowest µ s the stablephase is the thermal gas phase with n = 0. Above T c = T χ (0) only the symmetric phaseexists. Right: The quark number susceptibilities χ (continuous) and χ (dashed) per N c including the normalisation factor 4 / (45 π ). The limit of χ /T at large µ is , that of χ is2 /π . – 38 – igure 14 . The model Polyakov line plotted using (5.13) for µ = 0 , . , .
4. The red curvesare for the symmetric and the blue ones (at smaller T ) for the broken phase. The dashed lineindicates the discontinuity at T h . (4.7), gives F (cid:48) ( T, µ ) = − α (cid:48) κ ( λ h ) V eff ( λ h , τ h ) c s ( λ h , τ h ) . (5.13)On the RHS λ h , τ h are functions of T, µ on the LHS, for the symmetric phase τ h = 0.In the symmetric phase at large T , F (cid:48) ( T ) → − L UV2 α (cid:48) . (5.14)In the symmetric phase we shall, somewhat arbitrarily, fix the integration constantin the integration of F (cid:48) ( T ) so that F s ( T χ ( µ )) = 0. Normalising (cid:104) L (cid:105) to 1 at large T (actually, due to limitations of numerics, at T = 10 T c ) one obtains the red large T curves in Fig. 14 for µ = 0 , . , .
4. For the broken phase we shall enforce continuityat T χ ( µ ) by demanding that also F b ( T χ ( µ )) = 0. This is reasonable for the 2nd ordertransition which is the case here. The discontinuity shown for µ = 0 is due to thetransition at T h .It would be very valuable to derive a theoretically better founded gravity dual forthe Polyakov line. There is no order parameter for deconfinement but this operatoranyway is and will be used in lattice Monte Carlo studies. Now that we have two thermodynamic variables we have three second derivatives of p ( T, µ ). Out of the standard quantities C V and C p are complicated to compute in thepresent framework, but it so happens that the formula for the sound speed squared c s = dpd(cid:15) = s dT + n dµT ds + µ dn = b h [ T (cid:48) ( λ h ) + ˜ n π µ (cid:48) ( λ h )]3 b (cid:48) h ( λ h )( T + ˜ n π µ ) , (5.15)– 39 – = m= m= ê T c H L c s Figure 15 . Sound speed squared plotted vs
T /T χ at fixed µ . At small T and small µ soundvelocity drops markedly, the more the smaller µ is. At large T in the symmetric case c s atlarge T approaches 1 / where all quantities are to be taken at fixed ˜ n , can be directly evaluated. Ratherfortunately, our method of computation makes it trivial to take into account the extracondition among the fluctuations of pressure and energy density in (5.15), they are tobe taken at fixed n/s and, due to (4.17) this is just fixed ˜ n . In particular, for ˜ n = 0the derivative should be taken in the direction of T , as is usually done, though in thisdirection the volume density of entropy varies.The most interesting region is that near the phase transitions. Fig. 15 shows c s plotted vs T /T χ (0) (numerically T χ (0) = 0 . µ = 0 .
2, 0 . .
6, i.e. as onemoves vertically in the T direction in Fig. 11. At very large T , outside the figure, c s approaches the conformal value 1 /
3. For µ = 0 . c s approaches some fixed value at T = 0. T → limit The T → T = 0, can bereached by putting f = 1 in the metric Ansatz [32]. In this limit one can determine,for example, the lowest states of the mass spectrum of the theory. We have tried toapproach T = 0 by using the thermal Ansatz by trying to make T ∼ f (cid:48) ( z h ) as smallas possible. This clearly leads to large numerical fluctuations. The overall structure ofthe phase diagram along the T = 0 axis should nevertheless be as shown in Fig. 11,for the potentials used here. We are still, for example, missing the constant T and µ curves on the ( λ h , ˜ n )-plane in the chirally broken phase. With these one could morereliable conclude what p ( T → , µ ) is in both symmetric and broken phases.– 40 – Conclusions
We have analysed in this paper a non-fine-tuned gauge/gravity duality model for hotand dense QCD in the limit of large number of colors and flavors. The model contains5-dimensional gravity with AdS symmetry on the boundary, a scalar dilaton for con-finement and asymptotic freedom, a scalar tachyon for quark mass and condensate andthe zeroth component of a bulk 4-vector for chemical potential and quark number den-sity. The potentials of the model are constructed so that one obtains the correct QCDbeta function and mass running in the weak coupling region and color confinement inthe strong coupling region.Tuning the quark mass to zero, the main result of this paper is the phase diagramand a description of dynamical chiral symmetry breaking when temperature or densityis decreased. Chiral symmetry corresponds to solutions with vanishing tachyon, whichautomatically leads to the vanishing of both m q and the condensate. Spontaneouschiral symmetry breaking corresponds to solutions with non-zero tachyon, which areconstructed such that m q = 0 but nevertheless the condensate is nonvanishing. Byexplicit calculation of the pressures of chirally symmetric and broken phases we findthat the broken one dominates at small temperature and chemical potential, T < T χ ( µ ).The transition in between is of second order.When T is further decreased below T χ ( µ ), the system ultimately goes to anotherphase at some T h ( µ ) with a non-zero tachyon but without black holes: the thermal gasphase. We use this as a model for the low T hadron phase. In lattice Monte Carlosimulations one normally finds that the chiral transition (as identified by variation ofthe quark condensate) and deconfinement transition (as identified by the Polyakov lineor energy density discontinuities) coincide; there is effectively just one transition linebetween a quark-gluon plasma phase and a hadronic phase. Here we find that theselines are separated.The numerical effort needed to obtain the results presented here is extensive andwe have thus limited ourselves to a quantitative study of one set of potentials and thecase N f = N c . With improved numerical techniques many further questions can beaddressed. A set of potentials which describes all T = 0 QCD physics in quantitativedetail has been identified [32, 35]. Computing also its thermodynamics with goodaccuracy would make it possible to correlate zero and finite T properties reliably. Forexample, how does the requirement of linear Regge trajectories in particle spectra affectthe thermodynamics? How does the quark condensate behave as a function of density?Further, the approach to the conformal limit at x f = x c ≈ µ = 0 in [34] and it will be interesting to study also the T, µ phase diagram in thislimit. – 41 –here is a surprise in the phase diagram at low temperatures. Our analysis suggeststhat there is a new quantum critical regime with exotic properties at T = 0 whichrealizes the symmetries of the associated geometry, AdS × R . This exists both on the T = 0 segment of the chirality breaking plasma as well as the T = 0 line of the chirallysymmetric plasma.The presence of the AdS × R geometry in the holographic solution indicates thatthere is a scaling symmetry of the time direction which does not act in the spatial di-rections. Such symmetries have been called semilocal. This is an unexpected symmetryin a theory at finite density, but it is natural and generic in the holographic context[37] and appears even for simple black holes like the Reissner-Nordstr¨om black hole[38]. The physics in this critical regime is similar to that of a theory with zero speedof light: all spatial points decouple in the IR.The local RG pattern of such AdS solutions is fully compatible with the phasediagram we derived. It is an interesting question to determine physical implications ofthis scaling regime as well as its potential experimental signatures. We thank Blaise Gout´eraux, Misha Stephanov and Aleksi Vuorinen for discussions.This work was partially supported by European Union’s Seventh Framework Pro-gramme under grant agreements (FP7-REGPOT-2012-2013-1) No 316165, PIF-GA-2011-300984, the ERC Advanced Grant BSMOXFORD 228169, the EU program ThalesMIS 375734 and was also co-financed by the European Union (European Social Fund,ESF) and Greek national funds through the Operational Program “Education and Life-long Learning” of the National Strategic Reference Framework (NSRF) under “Fundingof proposals that have received a positive evaluation in the 3rd and 4th Call of ERCGrant Schemes”. TA thanks the Vaisala foundation for financial support. KT acknowl-edges financial support from the Academy of Finland project 267842.
A Fluctuation modes around the AdS point In this appendix we compute explicitly the amplitudes for the fluctuations of the AdS region discussed in Sec. 3.4. The amplitudes are easily obtained by solving the linear– 42 –ystem provided by a given α ∗ , and in general depend on one undetermined (but non-vanishing) amplitude and a choice of radial gauge which can be fixed via B . For theV-QCD AdS fixed point, the fluctuations are as follows: α ∗ = − D (cid:54) = 0 Φ = − E D + B ) and λ = τ = C = 0 (A.1)This mode corresponds to one sort of finite temperature perturbation to AdS . It isthe AdS black hole studied in [38]. α ∗ = − , D (cid:54) = 0 and B = − D and λ = τ = C = 0 (A.2)which corresponds to a shift in the chemical potentialΦ( r ) = µ + r (cid:18) E + Φ r (cid:19) (A.3)and a metric of the familiar formd s = − r L (cid:18) − D r (cid:19) d t + L d r r (cid:0) − D r (cid:1) + C d (cid:126)x (A.4)Again, this is a black hole in AdS , related to the one obtained from the α ∗ = − r → r (cid:48) = 12 ( D + ρ ) and t → t (cid:48) = 2 τ (A.5)the metric (A.4) becomesd s = − ρ L (cid:18) − D (cid:48) ρ (cid:19) d τ + L d ρ ρ (cid:16) − D (cid:48) ρ (cid:17) + C d (cid:126)x (A.6)where D (cid:48) = D , which is the metric implied by the α ∗ = − B (cid:48) = − D (cid:48) . Note the rescaling of the time coordinate in the transformation(A.5). Because black holes in AdS are coordinate equivalent to the vacuum AdS solution [52], the black hole (A.6) obtained via this coordinate change lives at a differenttemperature T than its parent solution (A.4), T ρ /T r = 2.– 43 – ∗ = 0These are marginal modes corresponding to rescalings of space and time. They aredescribed by C , D (cid:54) = 0 and Φ = E D and λ = τ = 0 (A.7)When C (cid:54) = 0, then the volume form on the R changes by a factor ofVol R ≈ Vol R (cid:18) C C (cid:19) (A.8)and when D (cid:54) = 0 then one obtains a shift in the time coordinate t → t (cid:48) = (cid:112) D t so d t (cid:48) ≈ (cid:18) D (cid:19) d t (A.9)These are the conjugate modes to the α ∗ = − α ∗ = 1The last universal mode is the irrelevant perturbation conjugate to the α ∗ = − fixed point. The amplitudes aresomewhat complicated, but take the form λ (cid:54) = 0 C = Γ D = ∆ Φ = (Γ + ∆ − B ) E τ = 0 (A.10)where Γ = 2 λ (16 + 3 L λ ∂ λ V ) w λ (cid:104) − E w ) ∂ λ w + L E w (cid:112) − E w ∂ λ V f (cid:105) (A.11)and ∆ = B − (cid:20) λ (cid:0)
16 + 3 L λ ∂ λ V (cid:1) λ Γ + 63Γ (cid:21) + E w Γ (A.12)That this mode interpolates between the IR and UV solutions is suggested by the factthat for λ (cid:54) = 0 the spatial part of the metric acquires a non-trivial radial dependenceas per (A.11).For the non-universal exponents, one finds a simple perturbation α ∗ = α τ τ (cid:54) = 0 D = 2 E Φ B = α τ E Φ and λ = C = 0 (A.13)This is gauge equivalent to a mode consisting of only a tachyon fluctuation.Finally, there exists a somewhat more complicated perturbation– 44 – ∗ = α λ λ (cid:54) = 0 Φ = E (cid:0) ∆ (cid:48) (1 + α λ ) − B (cid:1) D = ∆ (cid:48) and τ = C = 0 (A.14)where ∆ (cid:48) = 1 α λ B − λ (4 − E w ) ∂ λ w + L E w (cid:112) − E w ∂ λ V f α λ (2 + α λ ) w (A.15)where α λ is given by (3.39). This is a perturbation which couples the dilaton fluctua-tions to the metric fluctuations, leaving the spatial part of the metric unchanged.Note that the above expressions for the perturbation amplitudes hold for genericvalues of the constant scalars λ and τ . In the special case of the divergent tachyon,many of the amplitudes simplify as in this case E w = 1. To wit, (A.11) becomesΓ = λ (16 + 3 L λ ∂ λ V ) w λ ∂ λ w . (A.16) B Numerical solution of the equations of motion
B.1 Definitions
The purpose of this appendix is to clarify the technical details of the numerical solutionsto the equations of motion, the scaling properties of the results, and their dimensionalanalysis in the A = ln b -coordinates. This is essential for extracting the physics out ofthe numerics. All the details are built in the numerical code SolveFiniteTTachyons deposited in [39].In this treatment, the solutions in the A-coordinate system are considered primary,and the z-system is just an auxiliary coordinate system used to relate the results toknown holographic formulae. The treatment extends [34], but we shall rewrite it ex-plicitly in the form that the actual numerical code [39] uses, and keep each stage of theequations dimensionally consistent.We first define the notation: the fields q ( A ) , f ( A ) , λ ( A ) and τ ( A ) are thefields produced by numerical equation solving, expressed as a function of the coordinate A . These will be referred to as level 1 solutions. The level 1 coordinate A is the onein which the numerics is defined, and the horizon sits at A ,h = 0. L evel 2 solutionsare obtained after f -scaling (see next section) and level 3 solutions [34] are the finalones with the fields, observables and the coordinate in the units corresponding to thedesired UV boundary conditions.We shall consider at first only V-QCD at µ = 0, and then devote a separate sectionto the µ (cid:54) = 0 case. – 45 –e define the coordinate z by dAdz = e A q ( A ) , (B.1)with the boundary condition z ( A = ∞ ) = 0. Notice that this is defined with the finalscaled level 3 fields, and so we have precisely one system of z -coordinates, which wenever scale. B.2 The f -scaling We generally want the function f to asymptote to 1 in the UV ( z → A → ∞ ) in order to have the standard Minkowski coordinate system with c = 1 on theboundary. When the boundary conditions are set at the horizon, this is not in generalguaranteed. Fortunately the equations of motion are invariant under a combined scalingof f and q , such that if f , q are solutions, then also the pair f = f f (B.2) q = f scale q , (B.3)with no change to the other fields or the coordinate A = A , is a solution forany value of f scale , although with different boundary conditions. Choosing f scale =1 / (cid:112) f ( A = ∞ ) gives us the desired solution. From now on, fields and coordinateswith the subscript 2 denote the numerical solutions scaled in such a way. We will callthese the level 2 solutions.In [39], this scale factor appears as fscale and is explicitly used in generating thescaled solutions. The solutions produced by SolveFiniteTTachyons are level 1 in thisnotation, whereas
SolveAndScaleFiniteTTachyons produces functions that are level2 in this notation. The scaling itself is carried out in
ScaleSolution , which can alsobe used to convert the level 1 solutions produced by
SolveFiniteTTachyons to level2 solutions.The solution produced by this scaling no longer corresponds to the initial conditionsset in the numerics. Specifically, if the original equation solver was started with thecondition q ( A ,h ) = q ,h (B.4) f (cid:48) ( A ,h ) = f (cid:48) ,h (B.5)then the new solution corresponds to q ( A ,h ) = q ,h f scale (B.6) f (cid:48) ( A ,h ) = f (cid:48) ,h f , (B.7)– 46 –ith the initial conditions for the other fields unchanged.The code [39] sets the initial conditions f (cid:48) ,h = 1 (B.8) q ,h = − √ (cid:114) V g ( λ h ) − V f ( λ h , τ h ) (cid:113) ˜ n L A κ ( λ h ) V f ( λ h ,τ h ) , (B.9)where the first is chosen arbitrarily, since the magnitude of f is anyway set by f -scaling,and the second was derived in (4.7). The post scaling boundary condition then simplyis f (cid:48) ,h = f (B.10) q ,h = − √ f scale (cid:114) V g ( λ h ) − V f ( λ h , τ h ) (cid:113) ˜ n L A κ ( λ h ) V f ( λ h ,τ h ) , (B.11)with the rest of the fields unchanged. B.3 The Λ -scaling The UV -expansion (see (4.12) and Appendix A in [34]) is A = ˆ A + 1 b λ ( A ) + b b ln( b λ ( A )) + O ( λ ) , (B.12) λ ( z ) = − b ln( z Λ) + O (cid:18) ln( − ln( z Λ))ln( z Λ) (cid:19) , (B.13) A ( z ) = − ln z L UV + O (cid:18) z Λ) (cid:19) , (B.14)where ˆ A is a constant of integration. Here L UV is the asymptotic value of − q ( A ) atlarge A (or equivalently A ). Using these, we findˆ A = ln( L UV Λ) = lim A →∞ ( A − b λ ( A ) + b b ln( b λ ( A ))) . (B.15)Since we want to find a solution where Λ = Λ , we write this in the formˆ A = ˆ A − ln L UV Λ = ln(Λ / Λ ) = lim A →∞ ( A − ln( L UV Λ ) − b λ ( A ) + b b ln( b λ ( A )))(B.16) We present here already the formula with the chemical potential included, for µ = 0 set ˜ n = 0.See section B.5 for further explanation. – 47 –nd observe that the solution with the shifted coordinate A = A − ˆ A has the requiredasymptotics. Since the equations of motion are invariant with respect to shifts in A , thisis also a solution of the equations, although with different boundary conditions. Theseare the level 3 solutions. We further denote e ˆ A = Λ / Λ ≡ Λ scale . This is the factorthat appears in [39] as Λ scale , although in some places it is (inaccurately) denoted assimply as Λ. This factor is dimensionless.Using (B.16) converges somewhat slowly for practical purposes due to the O ( λ ) = O ( A − ) corrections. We speed up that convergence by considering ˆ A as a function of A max as given by (B.16), where A max is the limit up to which the numerical solutionhas been computed. From the numerical process, we know the derivatives of the fields,so we can compute ˆ A (cid:48) ( A max ) and derive the formulaˆ A = ˆ A ( A max ) − ˆ A (cid:48) ( A max ) λ ( A max ) λ (cid:48) ( A max ) , (B.17)which cancels the O ( λ ) corrections. The value of ˆ A computed by this method is re-turned by SolveAndScaleFiniteTTachyons , which uses
ScaleSolution to scale thelevel 1 solutions to level 2 and to derive Λ scale .We now write the transformation equations explicitly: A ( A ) = A − ˆ A, (B.18) A ( A ) = A + ˆ A, (B.19) A h = A ( A ,h ) = − ˆ A. (B.20)Especially note that the horizon value of, for example, λ ( A h ) = λ ( A ( A h )) = λ ( A h +ˆ A ) = λ ( − ˆ A + ˆ A ) = λ (0). In other words, the horizon value of any field h in thesolution with the correct asymptotics, is the same as the value of the original function h coming from the numerics, evaluated at A ,h = 0.The conformal factor of the metric appears in several physical observables. In level3 coordinates it is simply b ( A ) = e A = e A − ˆ A = e A Λ scale . (B.21)In addition, the derivatives of fields in the z -coordinate system often play a role, andwe observe that for example d ( f ( z )) dz = dAdz dfdA (cid:12)(cid:12)(cid:12) A = A ( z ) = e A q ( A ) dfdA (cid:12)(cid:12)(cid:12) A = A ( z ) = e A − ˆ A q ( A ) df dA (cid:12)(cid:12)(cid:12) A = A ( A ( z )) (B.22)= e A Λ scale q ( A ) dfdA (cid:12)(cid:12)(cid:12) A = A ( z )+ ˆ A (B.23)– 48 –nd especially at the horizon d ( f ( z )) dz (cid:12)(cid:12)(cid:12) z = z h = 1Λ scale q (0) dfdA (cid:12)(cid:12)(cid:12) A =0 . (B.24)An identical result holds for any field.Since it is possible in this way to eliminate the need to explicitly shift the fields,and thus the need to keep track of one extra variable, the actual numerical code [39]does precisely this. In the code A always refers to A = A , the horizon is always at A = 0, and the fields used to compute the physical observables are level 2.Since the equations of motion are invariant under shifts of A without any corre-sponding change in the fields, the initial conditions for the fields themselves at horizonwhen expressed in terms of the A -coordinates are not changed by this scaling. Therelation between A and z is what changes. Note however that once we introduce thegauge field Φ corresponding to a chemical potential, this changes since Φ explicitlybreaks this shift invariance. We will return to that later. B.4 Physical observables at µ = 0Using the previous results, we can work out the formulas for physical quantities usedin the code. The temperature is4 πT = − dfdz (cid:12)(cid:12)(cid:12) z = z h = − scale q ( A ) dfdA (cid:12)(cid:12)(cid:12) A =0 = − f (cid:48) ,h Λ scale q (0)= f scale √ scale (cid:118)(cid:117)(cid:117)(cid:116) V g ( λ h ) − V f ( λ h , τ h ) (cid:115) n L A κ ( λ h ) V f ( λ h , τ h ) , (B.25)where we used (B.24), (B.7) and (B.6) . In the code, T is returned by TemperatureFromSols .Note that both f scale and Λ scale are dimensionless, with the function q carrying one di-mension of length, giving the correct unit of 1/length = energy.Also note that from (B.6) one sees that the unit of length in q ultimately comesfrom the potential, which is proportional to 1 / L UV2 . This shows that Λ scale is thedimensionless factor which tells the relation between the 4D boundary units and theunits of the potential.The entropy density comes from4 G s = b ( A h ) = e A h = e − A = 1Λ . (B.26)This is returned in the code by s4G5FromSols . Note that b ( A h ) is dimensionless, sothe entropy density picks up its units from the gravitational constant G .– 49 –he quark mass is expressed as τ ( z ) / L U V = m q ( − ln(Λ z )) − γ /b z (1 + O (1 / ln z ))= − m q ( A − ln(Λ L UV )) − γ /b q ( A )Λ scale e − A (1 + O ( A − )) ⇒ m q = lim A →∞ L − V τ ( A ) e A ( A − ln(Λ L UV )) γ /b scale ≈ L − V τ ( A max ) e A max ( A max − ln(Λ L UV )) γ /b scale , (B.27)where A max is the maximum A to which the equations of motion have been solved.Except for the appearance of Λ scale , the shift between A and A is O ( A − ) = O ( A − )for large A .Similarly as with the determination of ˆ A , the A − corrections to m q are ratherlarge at easily reachable values of A max . As before, we can consider m q as a functionof A max and take its value at another point A b < A max . Using from the above that m q ( A ) = m q (1 + kA − ) for some unknown coefficient k , we can cancel the O ( A − )corrections: m q = m q ( A max ) A max − m q ( A b ) A b A max − A b (1 + O ( A − )) . (B.28)Since we know the derivatives of the fields from the numerical process, we can go furtherand take the limit A b → A max , yielding m q = m q ( A max ) + m (cid:48) q ( A max ) A max . (B.29)In practice the finite difference method of (B.28) is slightly more stable and convergesonly very slightly slower, and that method is therefore used in the code by default. Thefunction QuarkMass computes the mass from the solutions with this method.
B.5 Chemical potential
In (2.2), we introduce a zero component Φ of a gauge vector field in the bulk to modela chemical potential in the boundary theory. It turns out that the UV asymptotics arenot affected by this addition, and so we will want to do similar scalings as in the zerochemical potential case.However, the full structure of the solution with respect to scaling the UV-variablesdoes change, since there are new terms in the equations of motion, of the form f (cid:48)(cid:48) + (4 − q (cid:48) q ) f (cid:48) − V f L A κ e − A Φ (cid:48) (cid:113) fκq τ (cid:48) − κ q e − A L A Φ (cid:48) = 0 , (B.30)where we have written (2.33) without substituting the solution of the Φ equation ofmotion. – 50 –f we have a solution with subscripts 1, including Φ which has as yet undefinedtransformation properties, we have0 = f (cid:48)(cid:48) + (4 − q (cid:48) q ) f (cid:48) − V f L A κ e − A Φ (cid:48) (cid:113) f κq τ (cid:48) − κ q e − A L A Φ (cid:48) (B.31)= f − f (cid:48)(cid:48) + (4 − q (cid:48) q ) f − f (cid:48) − V f L A κ e − A Φ (cid:48) (cid:113) f κq τ (cid:48) − κ f q e − A L A Φ (cid:48) (B.32)= f − f (cid:48)(cid:48) + (4 − q (cid:48) q ) f − f (cid:48) − V f L A κ e − A − A Φ (cid:48) (cid:113) fκq τ (cid:48) − κ f q e − A − A L A Φ (cid:48) . (B.33)From this we conclude that if f , q , λ , τ , Φ solve the equations of motion with UVasymptotics corresponding to the level 1 fields, then the corresponding level 3 functionssolve the equations of motion with the correct UV asymptotics, if the function Φ isreplaced with Φ, such thatΦ( A ) = e − ˆ A f scale Φ ( A ) = f scale Λ scale Φ ( A ) . (B.34)It is apparent by inspection that the rest of the equations of motion are also invariantunder this substitution. We naturally call Φ( A ) a level 3 gauge field.The addition of a new field of course also adds a new pair of initial conditions. Thefundamental physical constraint (2.19) in A -coordinates requires that we set Φ h = 0,so the remaining initial condition is determined by the derivative of Φ at the horizon,Φ (cid:48) h . Now given a level 1 solution, corresponding to the initial condition Φ ,h , the scaledsolution clearly corresponds to the initial conditionΦ h = f scale Λ scale Φ ,h . (B.35)The field Φ is a cyclic coordinate: its equation of motion is ddA ∂L f ∂ Φ (cid:48) = ddA −L A e A q V f κ Φ (cid:48) (cid:113) fκq τ (cid:48) − κ e A q L A Φ (cid:48) = 0 , (B.36)which we can immediately integrate to the form (2.15): −L A e A V f κ Φ (cid:48) q (cid:113) fκq τ (cid:48) − κ e A q L A Φ (cid:48) = ˆ n. (B.37)Different values of ˆ n correspond to different initial conditions for the Φ field. Eval-uating this at the horizon for a given solution or a set of initial conditions gives us the– 51 –alue of ˆ n corresponding to that solution. Specifically, using the standard boundaryconditions for starting the numerics we have in terms of the level 1 solution −L A V f,h κ h Φ (cid:48) ,h q ,h (cid:114) − κ h q ,h L A Φ (cid:48) ,h = ˆ n . (B.38)On the other hand, applying known scaling properties of the fields to the lhs of thesame expression for the level 3 solution leads toˆ n = −L A e A h V f κ Φ (cid:48) h q h (cid:114) − κ h e Ah q h L A Φ (cid:48) h (B.39)= −L A V f,h κ h f scale Λ scale Φ (cid:48) ,h f scale q ,h (cid:113) − κ f Λ f Λ L A Φ ,h (B.40)= ˆ n Λ . (B.41)Since the level 3 solutions were the final ones, this gives us the scaling property of ˆ n .We can solve (B.37) to yield an explicit expression for Φ (cid:48) in terms of ˆ n and theother fields (for ˙Φ, see (2.16)): L A Φ (cid:48) ( A ) = − e A qκ (cid:118)(cid:117)(cid:117)(cid:116)(cid:18) f κq τ (cid:48) (cid:19) (cid:34) ˆ n L A ˆ n L A + e A V f κ (cid:35) . (B.42)Since Φ appears in the equations of motion always in the combination L A Φ (cid:48) , we couldentirely eliminate the choice of L A at this stage by rescaling ˆ n → ˆ n L A . Thereforewe can set L A = 1 without loss of generality in the numerics. Plugging the re-sulting formula into the equations of motion gives us the equations (2.31)-(2.34) onwhich [39] is based on. Solving the highest derivatives from those leads to the formin TachyonEquationsOfMotion . Since A ,h = 0, ˆ n matches with the scale invariantquantity ˜ n of (4.6): ˆ n = ˜ n = ˜ n . (B.43)With this substitution and using the scaling properties it is apparent that, oncethe equations have been solved and subjected to the f -scaling to yield λ , f , τ and q , we can write Φ (cid:48) as L A Φ (cid:48) ( A ) = − e A q ( A )Λ scale κ ( λ ( A )) (B.44) × (cid:115)(cid:18) f ( A ) κ ( λ ( A )) q ( A ) τ (cid:48) ( A ) (cid:19) (cid:20) ˜ n ˜ n + L A e A V f ( λ ( A ) , τ ( A )) κ ( λ ( A )) (cid:21) . – 52 –his function is returned in the code by APrimeFromSols . It is the fully scaled form,but expressed as a function of the coordinate A , which has not been shifted, i.e. itis in the same coordinate system as all the other functions returned by the code. It islevel 2 in the same sense as the rest of the level 2 functions: the coordinate system issuch that the horizon is at zero, but the units are such that it needs no further factorsof Λ scale . It can be used fully consistently with all the other output functions, but notethat if for some reason the coordinate system would be shifted again, Φ (cid:48) would then bescaled again according to (B.34). B.6 Physical observables for µ (cid:54) = 0When µ (cid:54) = 0 we immediately have two new physical observables. First there is thequark number density, ˆ n = ˜ n Λ , (B.45)returned in the code by nFromSols . The relation to the physical quark number density n is given in the text in Eq. (4.16).An interesting point is that the dependence on the actual numerical solution isprecisely the same as for the entropy density s in (B.26). Thus one has a physicalinterpretation for the input parameter ˜ n , it is simply ˜ n = 4 πn ( λ h ; ˜ n ) /s ( λ h ; ˜ n ), whereone also inserted the constants given in (4.17).The other observable is of course the chemical potential itself. The holographicformula for it is µ = lim A →∞ Φ( A ) , (B.46)for which we need to integrate (B.37). The correct boundary condition is that Φ( A h ) =0, yielding µ = (cid:90) ∞ A h Φ (cid:48) ( A ) dA = (cid:90) ∞ Φ (cid:48) ( A ) dA (B.47)This, and also the function Φ( A ), is returned in the code by AAndMuFromSols , with ∞ replaced by the upper limit A max of the range for which the equations have beensolved. In addition, the code uses L A = 1, but any other choice can be implementedby simply scaling n and µ . C Determination of τ h ( λ h ; m q ) The quark mass m q can be computed from the formulas presented in Appendix Bgiven the initial conditions at horizon. However, for computing physical results, we– 53 – igure 16 . T = T ( λ h ; ˜ n ) and µ = µ ( λ h ; ˜ n ) for tachyonless solu-tions for the PotILogMod potential with ¯ µ = − x f = 1 and for ˜ n =0 , , , , , , , , . , , . , . , , . , , . , . , .
25. For µ the smallest val-ues of ˜ n are 0 . ,
1. Note that T develops a minimum around λ h = 0 . n > .
5. The
T, µ derived from here is in Fig. 18.
Figure 17 . T = T (˜ n ; λ h ) and µ = µ (˜ n ; λ h ) for PotILogMod potential with ¯ µ = − x f = 1and for some values λ h . are interested rather in finding a class of solutions corresponding to a predeterminedvalue of m q , in this paper specifically m q = 0. This is in principle a simple problemof numerical function inversion, but it is complicated by the fact that the inverse ismultivalued and that computing values of m q ( λ h , ˜ n, τ h ) takes a considerable amount oftime (of the order of 1 second per point on a single core of an Intel i7 level processor).The main task involves determining τ h ( λ h , ˜ n ; m q ), given a fixed pair ( λ h , ˜ n ). Wewill denote m q ( τ h ) ≡ m q ( λ h , ˜ n, τ h ). As discussed in previous papers [33, 34], m q ( τ h )may have several zeroes, corresponding to different Efimov vacuums, of which the moststable is the one with the largest τ h . This means that it is not enough to find a zero,– 54 –ut rather we have to be able to bracket an interval containing the last zero before theasymptotic rise of m q ( τ h ) at large τ h , or alternatively deduce that no zero at finite τ h exists. In addition, since this needs to be done in at least tens of thousands of pointson the ( λ h , ˜ n ) -plane, the search must be fully automated and reliable enough to notneed manual checking of the solutions.The function τ hFromQuarkMass in [39] does this with a heuristic method that willbe briefly described here. We omit some details, for which we invite the interestedreader to look into the the code itself.1. First we find a point where the solution exists and m q <
0, starting from aninitial guess, by default τ h = 1. If the solution does not exist at all at the initialguess, τ h is multiplied by 2 to form a new guess. This is repeated until a point τ h, exist where the solution exists is found. After that, a point where m q ( τ h ) < , τ h, exist ]. We denote that point by τ h, min .If this point is not found in a predetermined number of bisections (default is 80),we conclude that a chiral symmetry breaking solution does not exist for this pair( λ h , ˜ n ).2. We look for τ h, max such that m q ( τ h, max ) > τ h until τ h, max is found.3. We now have two points τ h, min < τ h, max such that at least one root of m q liesbetween them. Starting a numerical root finder in this bracket with Brent’smethod would be guaranteed to find a root, but unfortunately there is no controlover which root. We need to start looking for zeros of m q in this interval. Thisis complicated by the fact that the distances between the zeros in τ h becomeexponentially larger toward increasing τ h . The heuristic we use determines aninitial step length by the formula ∆ τ h = τ h, min (( τ h, max τ h, min ) N −
1) (default N = 10 ).Then m q ( τ h,i ) is computed at points τ h, min + n ∆ τ h, min , n = 0 , ,
2, and we form theunique parabola passing through all of these points. The distance between its tworoots is used to provide a local estimate of the distances between zeroes, whichis used to determine a new step length ∆ τ h . We then compute m q at intervals of∆ τ h until we find a zero, that is, m q changes sign during a step.4. Once a zero is found, we update ∆ τ h to be the distance between τ h, min and thezero divided by small safety factor (default = 5).5. We continue to iterate with step length ∆ τ h , and whenever a new zero (a changeof sign in m q ) is found, we update the step length ∆ τ h to the distance betweenthe two latest zeroes divided by the safety factor. This iteration is terminated,– 55 –nd we take the last zero found, τ h, last , as the correct root, when the followingconditions hold simultaneously:(a) m q has not decreased from the last step, since we know that asymptotically m q grows.(b) m q > m q > k max( m q ( τ h ); τ h < τ h, last )), where k is a heuristically determinednumber, typically a few hundred. This is the main condition used to ensurethat the search goes on for long enough to reach the region of asymptoticgrowth.Once the iteration described above completes, we are reasonably confident that τ h, last and τ h, last − ∆ τ h bracket the largest zero, and simply use a standard root finderimplementing Brent’s method to find the precise location of that root. D Numerical results for T and µ As everywhere in this paper, numerical results in this appendix are all computed forthe potentials (2.6)-(2.8) with ¯ µ = − and for x f = 1.First, Fig. 16 plots T and µ as functions of λ h for fixed values of ˜ n for the tachy-onfree chirally symmetric solutions. In Fig. 17 the roles of λ h and ˜ n are interchanged.From these one then determines the two families of curves T = T ( µ ; ˜ n ) , T = T ( µ ; λ h ) , (D.1)which, when plotted on the T, µ plane, form a grid, see Fig. 18.Further, Figs. 19 and 20 show the same for the solutions with a nonzero tachyon.Several of the curves have numerical fluctuations. Note that there is a region near theorigin where broken phase solutions do not exist. Putting the symmetric phase andbroken phase grids together one obtains the grid in Fig. 21, on points of which thepressure p ( T, µ ) is numerically computed as discussed in Section 5.1.
E Thermodynamics along λ end The lower limit λ end (˜ n ) of the physical region of the tachyonic solutions on the ˜ n, λ h plane plays an important role in the thermodynamics. We shall here analyse its prop-erties.The lower limit λ end (˜ n ) arises when one tries to determine what values of τ h arepossible so that after integrating towards the boundary m q = 0 is obtained. One finds– 56 – igure 18 . Formation of the symmetric phase grid on the T, µ plane. The curves marked1 . , . , . . . , . λ h , those marked 1 , , . . . . ,
12 are those for constant˜ n . Figure 19 . Plots of T ( λ h ; ˜ n ) and µ ( λ h ; ˜ n ) in the broken phase as functions of λ h for valuesof ˜ n varying from 0 to 23 . .
5, see the physical region in Fig. 7. From these onederives curves of constant ˜ n on the ( T, µ ) plane (rightmost panel) for the broken phase.
Figure 20 . Plots of T ( λ h ; ˜ n ) and µ ( λ h ; ˜ n ) in the broken phase as functions of ˜ n for values of λ h varying from 1 . , see the physical region in Fig. 7. From these one derives curvesof constant λ h on the ( T, µ ) plane (rightmost panel) for the broken phase. that τ h = τ h ( λ h , m q = 0 , ˜ n ) is a monotonically growing function of λ h (see, e.g., Fig.5 of [34]) which starts at some λ h = λ end (˜ n ). As long as ˜ n (cid:46) igure 21 . The symmetric and broken phase grids. situation in which τ h ( λ end ) = 0 but if ˜ n (cid:38) τ h ( λ end ) >
0, (see Fig.22). Anaccurate plot of λ end (˜ n ) has been presented in Fig. 7, in Fig. 22 it is shown togetherwith the location of the deconfining transition. As discussed in Section 5.1, the chiraltransition takes place along λ end .The extreme situation is that when τ h is so large that V f decouples due to the e − aτ factor. The large ˜ n limit of (4.19) is then simply˜ n max = V g ( λ h ) κ ( λ h ) . (E.1) t h H l end H ñ L , ñ; m q = L ñ t h l p = H ñ L l end H ñ L ñ l h Figure 22 . Left: Behavior of τ h ( λ h , ˜ n, m q = 0) for as a function of ˜ n . Right: The physicalregion of tachyonic solutions λ h > λ end compared with the values of λ h at which the brokenphase pressure vanishes. This is where the deconfining transition takes place, the correspond-ing temperatures T h ( µ ) are plotted in Fig. 1. The chiral transition takes place along λ end (seeSection 5.1). – 58 –he upper limit of λ h moves to infinity and the large λ h limit of (E.1) is˜ n max = 23 .
99 + 12 . λ / h √ ln λ h + 1 + O (cid:18) λ h (cid:19) . (E.2)In the T = 0 limit it seems that s ∼ b h goes to a finite limit even though T = 0.In fact, s b varies only very little along the chiral equilibrium curve. F Large scale behavior
Since our model has asymptotic freedom built in it, we can at large T fit the magnitudeof the pressure to the ideal gas limit p = N c (cid:20) π (1 + x f ) T + x f µ T + π x f µ (cid:21) . (F.1) T H l end H ñ L , ñ L ñ T m H l end H ñ L , ñ L ñ m Figure 23 . Temperature and chemical potential along λ end (˜ n ). L H l end H ñ L , ñ L ñ L p H l end H ñ L , ñ L ñ m Figure 24 . The scale factor Λ = 1 /b h and the pressure along λ end (˜ n ). Note that s ∼ b h varies only very little along the curve. – 59 –arge T means small λ h and we thus want to compute the pressure p ( λ h , ˜ n ) at verysmall λ h and finite ˜ n in the approximation b ( z ) = L z , z h = e / ( b λ h ) ( b λ h ) b /b . (F.2)In this section, the limit λ h → λ h (equivalentto z h ) is often omitted. Thus here V g = 12, κ = 1 and, for x f = 1, L ≡ L UV =(1 + x f ) / = 1 .
401 and V f = x f W = 12(1 − / L ) = 5 . , L V f = 12 . , (cid:113) V g − V f = 10 . . (F.3)The pressure is obtained by first doing the pressure integral (5.6) along λ h at ˜ n = 0and then at fixed λ h the pressure integral (5.7) from ˜ n = 0 to some ˜ n . The former issimple and gives p ˜ n =0 ( λ h ) = L πG z h . (F.4)The latter becomes in the approximation (F.2)4 G p λ h (˜ n ) = L z h (cid:90) ˜ n d ˜ n (cid:20) T (cid:48) (˜ n ; λ h ) + 14 π ˜ n µ (cid:48) (˜ n ; λ h ) (cid:21) . (F.5)To evaluate this we must work out A ( z ) and µ = µ ( z h , ˜ n ) from (2.18) and T = T ( z h , ˜ n )from (2.37) in the approximation (F.2). Here z h is equivalent to λ h due to (F.2) andthe order of arguments is irrelevant.Noting that (cid:90) x du (cid:112) y u = x F ( , , , − x y ) (F.6)one finds z h L A µ = L V f F ( , , , − ˜ n L A V f ) ˜ n = L V f (cid:18) ˜ n − ˜ n L A V f + · · · (cid:19) (F.7)and z h πT = V eff (˜ n ) V eff (0) = 1 − L V f (cid:18)(cid:114) ˜ n L A V f − (cid:19) = 1 − L L A V f ˜ n + L L A V f ˜ n + · · · (F.8)– 60 –his form of T shows explicitly that T vanishes at ˜ n = L A (cid:113) V g − V f = 10 . L A , i.e.,at the physical region boundary in Fig. 7 (where L A = 1). By taking the ratio one seesthat µ/ ( πT ) is essentially determined by ˜ n so that it grows monotonically from 0 to ∞ at the physical region boundary.Inserting these exact forms to (F.5) and integrating one finds that4 G p λ h (˜ n ) = L πz h (cid:18) − z h + πT + ˜ n µ (cid:19) (F.9)so that the final total pressure in the limit of λ h →
0, ˜ n finite becomes p = p ˜ n =0 ( λ h ) + p λ h (˜ n ) = L G z h (cid:18) T + ˜ n π µ (cid:19) = s (cid:18) T + ˜ n π µ (cid:19) = ( T s + µn ) , (F.10)where we also used (4.17). This further implies that (cid:15) = T s − p + µn = 3 p in this UVcorner of parameter space. Note the mixed notation, p = p ( z h , ˜ n ) is given directly bythe above equations, but if we want p ( T, µ ) we must solve z h = z h ( T, µ ) and ˜ n = ˜ n ( T, µ )from the exact expressions (F.7) and (F.8).To compare with (F.1), consider first the limit T → ∞ , µ = constant. Taking theratio of (F.7) and (F.8), expanding in ˜ n and inverting the series one obtains˜ n = 2 V f L L A µπT (cid:20) L (1 − L V f ) L A µ π T + · · · (cid:21) . (F.11)and 1 z h = π T (cid:20) V f L L A µ π T + O ( µ T ) (cid:21) (F.12)Inserting this to (F.10) gives p = L πG (cid:20) ( πT ) + V f L A L µ ( πT ) + 16 (cid:18) V f L A L (cid:19) (cid:18) L V f (cid:19) µ + O ( µ T ) (cid:21) . (F.13)The two parameters G and L A can be fixed by the magnitudes of the T and µ T terms. Comparing the T terms of (F.1) and (F.13) gives first [34] L πG = N c x f π . (F.14)Using this the µ T terms agree if, inserting (F.3) and x f = 1, L A = L V f x f x f = 5 x f (1 + x f ) / x f ) / − ≈ . x f = 1) . (F.15)– 61 –he parameter γ then is1 γ = V f L A L = 15 x f x f = 3011 ( x f = 1) . (F.16)However, one can also determine L A requiring agreement with the µ term. The answeris L A = (cid:115) x f (1 + x f ) / x f ) / − x f ) / − ) ≈ . x f = 1) . (F.17)The values are automatically remarkably close also for other values of x f , for x f = 4(F.15) gives 1.667 and (F.17) 1.461. Thus both terms are reproduced almost correctlyand without further parameters, We thus have fitted that L A in (2.4) is very close toone.For completeness, the µ /T term in (F.13) is − γ (cid:18) − L V f + 81 L V f (cid:19) µ π T = − . µ π T . (F.18)Thus p starts falling below p idea , the non-expanded result is in Fig. 25.The comparison of (F.1) and (F.13) can also be carried out in the limit T → ∞ , µ/T = constant. Fixing the normalisation at µ = 0 we simply have pp ideal = 1 + π ˜ n ( µT ) µT (1 + π µ T + π µ T )( V eff (˜ n ( µT )) /V eff (0)) , (F.19) Figure 25 . Computed values of the ratio of the symmetric phase pressure integral (5.2) andthe ideal gas approximation (F.1) at T → ∞ and µ/T = fixed. The dashed line shows theasymptotic T (cid:28) µ limit 0 . – 62 –here ˜ n ( µT ) is to be determined by inverting the ratio of (F.7) and (F.8) numerically;the small- µ/T terms were given in (F.11). The result is plotted in Fig. 25. The valueof L A was fixed so that the ideal gas µ term was correctly reproduced. Now one seesthat the good agreement extends to large values of µ , at µ = 4 T the deviation is 3%.One can work out analytically the asymptotic limit at µ (cid:29) T which corresponds to˜ n → ˜ n max = L A (cid:113) V g − V f . It depends on V g = 12 and V f and its numerical valueis 0 . T = 0 [32, 35] finds that the holographic and perturbativeQCD results for the correlator of vector flavor currents agree in the UV if L A L W πG = N c π , (F.20)see Eq. (C.10) in [35] with w = L A κ = L A and W = V f /x f . This matches exactlywith the combination of (F.14) and (F.15). Note that both the pressure at large T, µ andthe vector correlator at large momentum depend only on the combination L A κ ( λ h = 0),here we have assumed κ (0) = 1. These quantities can be fixed separately using thescalar correlator. Combining the result in Eq. (C.21) of [35] and (F.15) one finds that κ (0) = 2 L L A = 16(1 + x f )15 x f (cid:20) (1 + x f ) / − (cid:21) = 2 . , ( x f = 1) . (F.21)This modified value of κ would affect the normalisation of τ and consequently that of thechiral condensate, but not the results in this article. As another example of comparisonsof weak coupling and holographic computations one may also compare this result withan analogous analysis of the finite temperature correlators of the energy momentumtensor in the UV [53]. Using the thermal normalisation (F.14), the holographic resultof the shear correlator is too small by a factor 4 / References [1] J. Kogut and M. Stephanov,
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