A UV expansions

On finite-temperature holographic QCD in the Veneziano limit

Abstract:

Holographic models in the universality class of QCD in the limit of large number of colors and massless fermion flavors, but constant ratio , are analyzed at finite temperature. The models contain a 5-dimensional metric and two scalars, a dilaton sourcing and a tachyon dual to . The phase structure on the plane is computed and various 1st order, 2nd order transitions and crossovers with their chiral symmetry properties are identified. For each , the temperature dependence of and the condensate is computed. In the simplest case, we find that for up to the critical there is a 1st order transition on which chiral symmetry is broken and the energy density jumps. In the conformal window , there is only a continuous crossover between two conformal phases. When approaching from below, , temperature scales approach zero as specified by Miransky scaling.

1

1 Introduction

QCD in the Veneziano limit [1],

 Nc→∞,Nf→∞,NfNc=xf  fixed,λ=g2\scriptsize YMNc  fixed, (1.1)

is expected to display a host of interesting and mostly non-perturbative phenomena, including:

• The “conformal window” with a nontrivial infrared (IR) fixed point, which extends from to smaller values of . The region has an IR fixed point while the theory is still weakly coupled, as was analyzed in [2] (see also [3]).

• It is expected that at a critical , the conformal window will end, and for , the theory will exhibit chiral symmetry breaking in the IR. This behavior is expected to persist down to . For the IR theory is a conformal field theory at strong coupling, that progressively becomes weak as .

• Near and below , there is the transition region to conventional QCD IR behavior. In this region the theory is expected to be “walking”: This means that the theory appears to be approaching the IR fixed point as the coupling evolves very slowly for many e-foldings of energies. But chiral symmetry breaking is nevertheless triggered and in the deep infrared the coupling diverges as in QCD. The slow evolution of the coupling has been correlated with a nontrivial dimension for the quark mass operator near two, rather than three (the free field value). IR observables are expected to obey the Miransky scaling [4, 5] as from below.

• New phenomena are expected to appear at finite density driven by strong coupling and the presence of quarks. These include color superconductivity [6, 7]. In this case, however, gauge invariant vevs are effectively double trace operators and the phase structure is determined at the next to leading order in .

The existence of the “walking” region makes the theory extremely interesting for applications in dynamical electroweak symmetry breaking (technicolor). This has also motivated an intensive lattice Monte Carlo work during recent years [8, 9, 10]. The bulk of this work has been done at zero temperature; recently there appeared the first attempts to go to finite for QCD with , up to 8 [11, 12, 10] and for non-QCD-like theories [13]. Chiral effective theories have also been applied [14, 15, 16, 17, 18, 19, 20, 21].

The aim of the present work is to study a class of holographic bottom-up models (V-QCD) that belong to the universality class of QCD with massless quarks in the Veneziano limit [22] at finite temperature and zero chemical potential. We will calculate the temperature dependence of the free energy density (= pressure = ) and of the quark condensate (). The former acts as an effective order parameter for deconfinement (at large ), for which there is no true order parameter associated with a symmetry.2 The quark condensate is a true order parameter for chiral symmetry if quarks are massless. The calculation is carried out for the full range of , .

Discontinuities or rapid variations in pressure (or energy density) and quark condensate can be used to define phase boundaries associated with deconfinement and chiral symmetry restoration temperatures and . We will use the usual nomenclature: If the th derivative of is discontinuous, the transition is of th order. We also consider continuous crossovers which are identified by using the scaled quantity . Its maximum defines the crossover temperature . The phase diagram is defined as a plot of all phase boundaries on the plane. The phase diagrams we present will also contain a rich structure of metastable states, namely local (but globally subleading) minima of the free energy.

In the holographic approach the thermal transitions will be transitions between various 5-dimensional black hole and “thermal gas” metrics and the nomenclature of transitions, explained later in great detail, will be correspondingly different. The holographic approach is constrained but not fully constrained and we cannot give a precise prediction of the phase diagram of hot V-QCD. We can state the most plausible behavior but we can also mention a few other alternatives. We will always find the analogues of and , but we will also find transitions with no obvious QCD interpretation. Whether these reflect real physics of hot QCD in the Veneziano limit or whether they are artifacts of the holographic approach will be an interesting problem for further study.

The usual expectation is that there is a 1st order line at ; in the large limit one can actually prove that [14, 15]. The main class of our predictions reflect these properties: for smaller we find that deconfinement and chiral symmetry restoration coincide, but for approaching the deconfining and chiral transitions can become separate so that (see, for example, Fig. 13 below). The chiral transition is then of 2nd order (and mean field type). Furthermore, for smaller the separate 2nd order chiral transition is in the metastable region so that it can be reached if the system is supercooled [23]. One might here add that for stable phases may be reached at large chemical potential [24, 25].

The starting point of our finite temperature analysis is the holographic model introduced in [22], based on previous theoretical ideas in [26, 27, 28, 29, 30]. Moving to finite implies studying black hole solutions of the action in [22]. A defining characteristic of this class of models is that they contain full backreaction between the duals of the color and flavor degrees of freedom. Earlier work [31, 32, 33, 34] on thermodynamics in such bottom-up models imposed quasiconformality directly on the beta function of the theory. One should note that walking behavior and the related “conformal transition” at have also been studied in top-down models [35, 36, 37, 38], as well as in simpler bottom-up models [39, 40] which do not attempt to model the backreaction. See also the review [41] on introducing backreacted flavor in the top-down models.

In this introduction we will first describe the special properties of V-QCD from [22] and then discuss general properties of its black hole solutions. Section 2 will contain a detailed discussion of the action of the model and of the two characteristic classes of scalar potentials. Section 3 presents the Einstein equations of the model, describes how they are numerically solved and, finally, how thermodynamics is computed from the numerical bulk fields. A particularly delicate issue here is the fixing of the quark mass to zero. We also briefly comment on fixing to some nonzero value. An extensive list of numerical results is given in Section 4. From these, the types of phase transition lines the models predict are determined. In Section 5, techniques for computing the condensate are described and several numerical results are given. One should note that this, as well as many other numerical issues in the model, are technically very demanding. Finally, Section 6 contains a discussion of what are the effects of making the quark mass nonzero. Several detailed considerations are collected in Appendices.

1.1 V-QCD at zero temperature

In [22] a class of bottom-up holographic models was introduced (named V-QCD) and shown to be in the universality class of QCD in the Veneziano limit at zero temperature and density. These were 5-dimensional models of two scalars coupled to gravity. One of the scalars, the “dilaton” , is dual to (the QCD gauge coupling constant, or more precisely the ’t Hooft coupling). The other scalar, the “tachyon” , is dual to the quark mass operator . The potentials and interactions were modeled along successful bottom-up models for YM, namely Improved Holographic QCD (IHQCD) [26, 27, 28] and the idea that string theory tachyon condensation describes chiral symmetry breaking [29, 30, 42].

The bulk action considered was

 S=Sg+Sf,Sg=M3N2c∫d5x√−g[R−43(∂λ)2λ2+Vg(λ)], (1.2)

with the ’t Hooft coupling (exponential of the dilaton ) and the tachyon3 action4

 Sf=−xfM3N2c∫d5x Vf(λ,τ)√det(gμν+κ(λ)∂μτ∂ντ). (1.3)

The pure glue potential has been determined from previous studies [27]. The tachyon potential must satisfy some basic properties determined by the dual theory or by general properties of tachyons in string theory: (a) To provide the proper dimension for the dual operator near the boundary (b) To exponentially vanish for . The function captures, among other things, the transformation from the string frame to the Einstein frame in five dimensions. The class of potentials that were investigated in [22] are of the form

 Vf(λ,τ)=V0f(λ)e−a(λ)τ2. (1.4)

In the Veneziano limit, the back-reaction of the flavor sector on the glue sector is fully included.

As with IHQCD, it was arranged that the theory is asymptotically AdS in the UV up to logarithmic corrections in the bulk coordinate. The function is such that the potential , when the tachyon has not condensed () has an extremum 5 at a finite value . As we approach the Banks-Zaks region [2], , the value of approaches zero. Without the tachyon, , the equations of motion imply that also , i.e., is an IR fixed point. When the dynamics of is included, the system approaches but is driven away from it as long as (see Fig.7 of [22]).

The dimension of the chiral condensate was calculated in the IR fixed point theory from the bulk equations. It was found that it decreases monotonically with for reasonably chosen potentials. It crossed the value 2 at where corresponds to the end of the conformal window as argued in [43].

The lower edge of the conformal window lies in the vicinity of 4. Requiring the holographic -functions to match with QCD in the UV, we find that

 3.7≲xc≲4.2, (1.5)

where the bounds are not strict but hold approximately for potentials that have smooth -dependence in the UV.

There is also a phenomenological heuristic argument for the value , simply from counting degrees of freedom. At low chiral symmetry is broken and the massless degrees of freedom are Goldstone bosons. At large there are weakly coupled degrees of freedom. These numbers are equal for . Conformal window and the location of its edge was also discussed within holographic frameworks related to V-QCD in [44, 33].

Apart from , there is a single parameter in the theory, namely where is the UV value of the (flavor independent) quark mass. For each value of , the bulk equations were solved with fixed sources corresponding to fixed . The vevs were determined such that the solution is “regular” in the IR. The notion of regularity is tricky even in the case of IHQCD (pure glue), as there is a naked singularity in the far IR. For the dilaton this has been settled in [26, 27]. For the tachyon the notion of regularity is different and has been studied in detail in [30].

The regularity condition was implemented in the IR. After solving the equations from the IR to the UV (this was done mostly numerically), there is a single parameter that determines the solutions as well as the UV coupling constants and vevs, and this is a real number controlling the value of the tachyon in the IR. This number reflects the single dimensionless parameter of the theory.

For different values of and the following qualitatively different regions were found in [22]:

• When and , the theory flows to an IR fixed point. The IR conformal field theory is weakly coupled near and strongly coupled in the vicinity of . Chiral symmetry is unbroken in this regime (this is known as the conformal window).

• When and , the tachyon has a nontrivial profile, and there is a single solution with the given source, which is “regular” in the IR. The IR theory is a theory with a mass gap.

• When and , there is an infinite number of regular solutions with nontrivial tachyon profile, and a special solution with an identically vanishing tachyon and a nontrivial IR fixed point. The infinite number of solutions with nontrivial tachyon are classified by their number of zeros. The solution with the lowest free energy is the one with no zeros.

• When and , the theory has vacua with nontrivial profile for the tachyon. For every non-zero , there is a finite number of regular solutions that grows as approaches zero.

In [22] two large classes of tachyon potentials were identified. Potentials in class I, have constant in (1.4). In this case the tachyon diverges exponentially in the IR for the regular solution

 τ∼r→∞τ0exp[Cr], (1.6)

where is a known constant (see Appendix B) and is the only integration constant controlling the solution. It determines the source (mass) in the UV. Potentials in class II, have as , and a tachyon that diverges in a milder way in the IR as

 τ∼r→∞C√r−r1, (1.7)

where again is known and is the single integration constant controlling the regular solution. The qualitative conclusions above and below were valid for both classes of potentials.

In the region where several solutions exist, there is a interesting relation between the IR value controlling the regular solutions, and the UV parameters, namely . This is determined numerically, and a relevant plot describing the relation between and at fixed is in Fig. 1.

The solutions are characterized by the number of times the tachyon field changes sign as it evolves from the UV to the IR. For all values of there is a single solution with no tachyon zeroes. In addition, for each positive there are two solutions6 which exist within a finite range , where the limiting value decreases with increasing , and one solution for . In particular, for large enough fixed , we find that only the solution without tachyon zeroes exists.

For , out of all regular solutions, the “first” one without tachyon zeroes has the smallest free energy. The same is true for , namely the solution with nontrivial tachyon without zeroes is energetically favored over the solutions with positive as well as over the special solution with identically vanishing tachyon, which appears only for and would leave chiral symmetry unbroken. Therefore, chiral symmetry is broken for .

In the region just below , [22] found Miransky scaling for the chiral condensate. As ,

 Missing or unrecognized delimiter for \left (1.8)

For , let be the mass of the tachyon at the IR fixed point and the IR AdS radius. The coefficient is then fixed as

 ^K=π√ddx[m2\scriptsize IRℓ2\scriptsize IR]x=xc . (1.9)

The behavior at and below the conformal transition at is to a large extent independent of the details of the model. In particular, no information on the nonlinear terms in the tachyon EoM is needed or how the IR boundary conditions are fixed. In the same region, “walking” of gauge coupling is realized. The YM coupling flows from small values to values very near , remains approximately constant for many e-foldings of energy (in this regime the tachyon remains small), and then runs off to infinity, driven by a large value of the tachyon field in the IR. The walking is related to a long section of the solution which is similar to the one studied in earlier bottom-up models for walking [39].

The finite temperature analysis of V-QCD amounts to studying all black hole solutions with appropriate boundary conditions. To start with, any zero temperature solution becomes a candidate saddle point at finite temperature by compactifying time on a circle of radius . Any other competing black hole solution must have the same boundary conditions as well as a regular horizon in the IR.

As the dilaton always has a nontrivial UV source, it will always have a nontrivial profile in the black-hole solutions. With the tachyon, things can be different. In the massless case, its source is zero. Therefore there are two possible options (as in the zero temperature configurations discussed above): either it is identically zero (if the vev is also zero) or it is non-zero (implying a non-zero vev).

Therefore we have two large classes of black holes in the massless case: those with and those with . We will first consider the tachyon-free class.

1.2 Black holes without tachyon hair

If , we have black holes in a single scalar theory, with potential from (1.4). This is a potential with an extremum for 7 and no extremum when .

Black hole solutions for such potentials were discussed in generality in [27]. After fixing all invariances, they are characterised by a single IR constant, , the value of the dilaton at the horizon. The plot relating the temperature to contains important information about the thermodynamics of such black holes. Small values of denote large black-holes whereas larger values of correspond to smaller black holes (smaller horizon size and entropy). In all plots of this paper, dilatonic black holes without tachyon hair are denoted by red lines in the respective -diagrams, and we shall call the corresponding function .

When , can become arbitrarily large at zero temperature, implying that can also be arbitrarily large for the finite temperature configurations. At finite temperature there are two branches: large black holes which are stable and small black holes which are unstable. If the black-hole thermodynamics is stable, otherwise it is unstable. There is a minimum temperature above which such black holes exist as shown, for example, by the black line in Fig. 22 (left or right).

When , we have two possibilities. The first is that the potential has an extremum at for all , with and . The second is that such extremum only exists for , where . We shall denote these potentials with a star subscript.

At finite temperature, and when the potential has no extremum, the black hole without the tachyon hair exists for all positive . For the potentials studied here, function is qualitatively similar to the YM case () [27]. As shown in Fig. 17 (top-left) and in Fig. 19 (left), there are two black hole branches, which exist above some minimum temperature. The branch at low is thermodynamically stable, while the large- branch is unstable.

When the extremum is present, . The temperature of the black-hole corresponding to is , while that of has . There is no minimum temperature here. For any temperature there is always at least one black-hole solution. There are several possibilities that are shown as red lines in Figs. 7 (left), 9 (top), 10 (left) and 12 (left).

When is large, but still smaller than , the relation is one-to-one but contains a bump (a change of concavity) like in Fig. 9 (top). Then this is accompanied by a crossover behavior, signaled by a bump in the trace of the stress tensor , (aka interaction measure) as shown in Fig. 9 (bottom-right).

At low enough , the relation is not always one-to-one, as can be seen in Fig. 10 (left) or in Fig. 22. Then there are points where . In such a case there can be a first order transition between the stable branches of the black hole solutions. This is a remnant of the deconfining transition at (pure YM). In Fig. 22 both left and right several curves in the -plane for different indicate the successive structure of dilaton black holes (red lines). The black line corresponds to the pure YM () limit.

When we are in the conformal window. The only black holes that exist here are those without tachyon hair. The relation is monotonic and there is a continuous transition to the black-hole phase at , as in the AdS case in the Poincaré patch. The thermodynamic functions, especially the interaction measure, show a crossover maximum at a temperature that is moving towards the UV as .

1.3 Black holes with tachyon hair and zero quark mass

When we have black holes in the two scalar theory. The tachyon starts as near the UV boundary as the source (quark mass) vanishes. In all plots of this paper, such black holes (with both dilaton and tachyon hair) are denoted by blue lines in the respective -diagrams, and we shall denote the corresponding functions by . They are still one parameter solutions and can be parametrized again by the value of at the horizon, which translates into the temperature. These black holes usually exist for all and our discussion below focuses in this region.

Because the presence of the nontrivial tachyon perturbs and annuls the possible nontrivial IR fixed point, for such black-holes, can take arbitrarily large values. This can be seen from the blue lines in Figs. 7 (left), 9 (top), 10 (left) and 12 (left). For all such black holes, the chiral condensate is determined by the regularity of the black hole solution. It decreases as decreases, and at some point it vanishes. At this point, the blue line in the ()-diagram merges with the red line corresponding to a that we call throughout the paper. This can be seen in all the figures mentioned above.

The shape of the blue line can vary as a function of and the type of potential. There are three typical examples of shapes:

• Simple lines that are monotonic as the one depicted in Fig. 12 (left). This is an example of a monotonic blue branch where all such black-holes are thermodynamically unstable. Moreover, they have a minimum temperature. In such a case, they can never be thermodynamically dominant. At some temperature the vacuum thermal solution is dominated by a dilaton black hole on the red line, and the chiral restoration transition is 1st order.

• Lines with two branches as the one depicted in Fig. 10 (left). Here the blue line has two parts one (to the left) that is thermodynamically stable and another to the right that is thermodynamically unstable. In such a case, the system is in the thermal vacuum solution at low enough temperatures, then jumps with a 1st order transition to the tachyon-hairy solution (the part of the blue line that is thick in Fig. 10 (left)) which still break chiral symmetry, and then eventually smoothly transits to the red line at the point where the blue and red lines merge, via a chirally-restoring 2nd order transition.8

• Lines with more than two branches as the one depicted in Fig. 11 (left). In this example the blue line has four branches, two unstable and two stable. There are in total three phase transitions here, first from the vacuum thermal solution to the rightmost blue thick branch, then to the intermediate thick blue branch and finally a 2nd order chirally restoring transition to the red branch at the point they touch. In this case there are two 1st order transitions between three chirally breaking phases, and a 2nd order one to the chirally symmetric phase.

A concrete overall view of the dependence is presented in Fig. 2, in which is plotted for potentials of type II with SB normalisation (definitions specified later) for various . One sees clearly how the pure (black) YM curve is approached for . The thick curves represent stable phases; when a thick curve ends, the system makes a 1st order transition to the low phase. When thick curves change from red to blue curves, a 2nd order transition to a chirally broken phase takes place. For a more accurate picture of small , see Fig. 22.

1.4 The phase structure of different V-QCD models

There are three main ingredients that characterize a priori different QCD models which, however, have the same phase structure and qualitative behavior at zero temperature:

• The asymptotics of the tachyon solution in the IR. This is controlled by the behavior of the function in the tachyon potential in (1.4). When is constant, the tachyon diverges exponentially in the IR, and we call such potentials of type I. When diverges as in the IR ( large) then the tachyon diverges as a square root in the IR, and we call such potentials of type II.

• For any potential, the UV constant factor of in (1.4), defined in (2.22) can vary in finite range, which in appropriate units is , as in (2.30). We pick for each type of potential three indicative values of that in general might give different physics, namely , , and .9 We also consider -dependent value, specified in (2.32) that corresponds to the normalization of the UV degrees of freedom of the free energy to the Stefan-Boltzmann limit of QCD.

• A final variation can be obtained on all of the above by using a glue potential in (1.4) that has

(a) an extremum for all in the appropriate range, .

(b) an extremum only in part of this range, . We will denote the potentials in this case by a star subscript.

According to the above options PotI denotes a potential in the type I class, with and an IR critical point that exists only down to a finite .

Let us then summarize the phase structure of the model as and the temperature are varied (at zero quark mass). In general one expects the phase diagram of Fig. 8, so that for there is the 1st order transition at finite temperature, which also separates the chirally symmetric and broken phases. For the low temperature and high temperature configurations correspond to a tachyonless black holes, and, one expects a continuous crossover between these two.

For the various potentials presented above, this phase diagram is indeed obtained in the zeroth approximation, but for there are additional details which depend on the choice of potentials as follows.

• For potentials I the phase structure depends strongly on the choice for (see Fig. 18). For the lowest value , there is only one 1st order transition at10 for all , except possibly for very close to , where solving the phase diagram numerically becomes demanding. As is increased, a complicated structure appears near , where we have two 1st order transitions between chirally broken phases, and the restoration of chiral symmetry at a 2nd order transition at even higher temperature. At even higher the 1st order transitions combine again into a single one, but the separate 2nd order transition continues to exist for close to . At low , there is also a surprising change as increases. The chiral symmetry breaking phases disappear, but there is a 1st order transition between two chirally symmetric black hole phases at a finite temperature instead.

• For potentials II the dependence on is milder (see Figs. 13 – 16). At high , for low up to some value , there is only the 1st order transition at11 . When , the chiral symmetry restoration takes again place at a 2nd order transition at such that . For decreasing , increases, and finally disappears by joining with .

• For the potentials I, the phase structure is the standard one for high , i.e., a 2nd order transition and a 1st order one with critical temperatures within a range , with the former separating the chirally symmetric and broken phases (see Fig. 19). For lower there is only one 1st order transition. For , in the region where the effective potential does not admit an extremum, chiral symmetry is intact at all temperatures. We find a single 1st order transition between chirally symmetric thermal gas and black hole phases.

• For potentials II, the phase structure is simple (see Fig. 17): there is a single 1st order transition for all . In particular, the system is in a chirally broken phase at low temperatures, even in the region of low where the effective potential does not have an extremum.

2 Defining V-QCD

2.1 Gravity action of the model

The action of V-QCD is [22]

 S=M3N2c∫d5xL≡116πG5∫d5xL, (2.10)

where12

 L = [√−g(R−43(∂λ)2λ2+Vg(λ))−Vf(λ,τ)√det(gab+κ(λ,τ)∂aτ∂bτ)] (2.11) = √−g[R+[−43gμν∂μϕ∂νϕ+Vg(λ)]−Vf(λ,τ)√1+grrκ(λ(r))τ′(r)2].

The metric Ansatz is

 ds2=b2(r)[−f(r)dt2+dx2+dr2f(r)],b(r)=eA(r)∼r→0L\scriptsize UVr, (2.12)

and the two scalar functions, sourcing and sourcing , are

 λ=λ(r)=eϕ(r),τ=τ(r). (2.13)

In the second form has been factored out of the DBI action. The Gibbons-Hawking counter term would be

 SGH=−∫d4x√−γ[2K+6L% \scriptsize UV+L\scriptsize UV2R(γ)], (2.14)

with, for a hypersurface const,

 K=√f2b(8b′(r)b+f′(r)f). (2.15)

Notice also that we have set the gauge fields , which are dual to the left and right handed fermion currents, to zero, and neglected the Wess-Zumino terms. These terms do not affect the thermodynamics of the models.

The background solution of the dilaton and the warp factor are identified as the ’t Hooft coupling and the logarithm of the energy scale of the dual field theory, respectively [26]. As a matter of convention, we shall fix the normalisation of so that its relation to the perturbative QCD coupling is

 λ(r)=g2(b(r))8π2. (2.16)

The results of the model are independent of this normalisation, changing one simply has to change the potentials by . The convention of [22], for example, is obtained by shifting by .

Important ingredients of the model are the relation of the bulk fields at to the QCD beta and quark mass anomalous dimension functions evaluated for a coupling at scale . Motivated by the connection to field theory, one defines

 β=dλdb/b=λ′(A)=−b0λ2−b1λ3−b2λ4…,γ=τ′(A). (2.17)

Matching with the perturbative expansion of the QCD beta function gives

 b0=13(11−2xf),b1=16(34−13xf). (2.18)

The action contains the gluonic potential and the fermionic potential , which will be specified to the form

 Vf(λ,τ)=xfVf0(λ)e−a(λ)τ2 . (2.19)

The detailed form of these and the functions will be discussed in the following subsections.

2.2 Construction of the potentials

The potentials can be constructed in stages. First one fixes the potentials and up to order in the UV, using the two scheme independent coefficients of the beta function. This analysis is simplified by the fact that the tachyon decouples in the UV. Next one fixes the UV behavior of the functions and , which parametrize the tachyon dependence of the action using the similarly scheme independent UV running properties of the quark mass and the condensate. Finally, one fixes the large behavior of the potentials by requiring that the model reproduces known features of QCD in the IR, such as confinement, linear Regge trajectories, and reasonable zero-temperature phase structure. We shall discuss the various steps in detail below (see also [22]).

The potentials from the beta function in the UV

In the UV, since the tachyon vanishes much faster than the dilaton, we can first set it to zero. Then the dilaton profile can be linked to the effective potential [22] by using Einstein’s equations [26]. Defining , to order ,

 Vg−xfV0f = 12L2\scriptsize UVexp[−89∫λ0dλβλ2](1−β29λ2) (2.20) = 12L2\scriptsize UV[1+89b0λ+(2381b20+49b1)λ2] (2.21) = V0−xfW0+(V1−xfW1)λ+(V2−xfW2)λ2, (2.22)

where we expanded

 Vg=V0+V1λ+V2λ2+O(λ3),Vf0=W0+W1λ+W2λ2+O(λ3), (2.23)

and where we have introduced an dependent AdS radius

 L\scriptsize UV=L(xf). (2.24)

Applying equation (2.21) for we have for the gluonic potential

 Vg = 12L20(1+89b% \scriptsize YM0λ+23(b\scriptsize YM0)2+36b\scriptsize YM181λ2) (2.25) = 12L20(1+8827λ+4619729λ2), (2.26)

where are the values of for and . In practice, one usually sets the (dimensionful) quantity .

By using equations (2.21) and (2.22) one can now solve for the coefficients of the fermionic potential:

 xfL20W0=12(1−L20L2% \scriptsize UV), (2.27)
 xfL20W1=323(b\scriptsize YM% 0−b0L20L2\scriptsize UV)=12⋅827[11−(11−2xf)L20L2\scriptsize UV], (2.28)
 xfL20W2 = 1281[23(b\scriptsize YM0)2+36b\scriptsize YM1−(23b20+36b1)L20L2\scriptsize UV] (2.29) = 12729[4619−(4619−1714xf+92x2f)L20L2\scriptsize UV].

These equations still involve one free parameter, which can be taken to be either or . We shall study two ways to fix this parameter. First, we can take to be constant. In this case [22]

 0≤L20W0≤2411, (2.30)

and the -dependent AdS radius is given by

 L\scriptsize UV=L0√1−112L20W0⋅xf. (2.31)

Second, we can make a special -dependent choice, which (as we shall show later) automatically normalises the free energy at large to Stefan-Boltzmann:

 L\scriptsize UV=L0(1+74xf)1/3. (2.32)

Further, we have to fix the dependence of the functions and in the tachyon part

 xfVf0(λ)e−a(λ)τ2√1+grrκ(λ(r))˙τ2, (2.33)

of the action, where . The leading logarithmic term to the UV expansion of the tachyon should be (remember that the energy dimension of is )

 τ(r)/L\scriptsize UV=mr(−lnΛr)−γ0b0=mr(−lnΛr)−32b0 (2.34)

to satisfy the scheme independent UV running of the quark mass. Here is the leading coefficient of the anomalous dimension of the quark mass in QCD, . By using the tachyon equation of motion one sees that this requires that for small ,

 κ(λ)a(λ)=23L2\scriptsize UV% [1−(89b0+1)λ+λ2+⋯]. (2.35)

Large λ behavior of the potentials

To specify the full potential we have to continue the small expansions to large . The guideline is quark confinement and chiral symmetry breaking at small and the appearance of an infrared fixed point at some (see [22]). Since there is no unique path to the result, we present the final forms of the potentials we use and motivate them.

We use the gluonic potential

 Vg(λ)=12L20[1+88λ27+4619λ2729√1+ln(1+λ)(1+λ)2/3] (2.36)

which is constructed from the expansion (2.25) by simply multiplying the term by the confinement factor

 √1+ln(1+λ)(1+λ)2/3 . (2.37)

Then has the proper large- behavior [26] but the small- behavior is left intact. One could add scale factors of type containing more parameters.

For the fermionic potential in

 Vf(λ,τ)=xfVf0(λ)e−a(λ)τ2 (2.38)

we consider two different choices. The first one is obtained directly using (2.27)-(2.29)

 Vf0 = 12L2\scriptsize UVxf[L2\scriptsize UVL20−1+827(11L2\scriptsize UVL20−11+2xf)λ (2.39) +1729(4619L2\scriptsize UVL20−4619+1714xf−92x2f)λ2].

Here one could as well use the parameter which is related to by

 L20L2\scriptsize UV=1−xfL20W012. (2.40)

For this choice the effective potential

 V\scriptsize eff(λ)=Vg(λ)−xfVf0(λ) (2.41)

has a single maximum at finite positive for all , indicating a (possible) infra-red fixed point.

The second choice is obtained introducing the confinement factor (2.37) also for the fermionic potential, i.e.,

 Vf0 = 12L2\scriptsize UVxf[L2\scriptsize UVL20−1+827(11L2\scriptsize UVL20−11+2xf)λ (2.42) +1729(4619L2\scriptsize UVL20−4619+1714xf−92x2f)λ2√1+ln(1+λ)(1+λ)2/3].

Now the effective potential has a maximum only at large . To see this concretely, consider again the case (2.32). The asymptotic large- behavior of now is times the function

 18476243−44619(1+74xf)2/3−4619+1714xf−92x2f243(1+74xf)2/3. (2.43)

This function is positive for small , negative at large () and has a zero at . Thus there is a (possible) fixed point only for .

Let us then discuss the IR behavior of the potentials and which appear in the tachyon DBI action. For the function we will consider the large- asymptotics

 κ(λ)∼λ→∞λ−4/3 . (2.44)

This is motivated by the fact that in the action the combination has the same asymptotics as at large , where is the metric factor in the string frame. To ensure that the fractional exponent limit at large does not spoil analyticity at small , we replace by in the expression for .

More precisely, two qualitatively different, acceptable choices for the IR asymptotics of (and ) were identified in [22]. These are produced by the following two choices. The first choice has

 a(λ)=321L2UV,κ(λ)=1[1+34(89b0+1)λ]4/3=1(1+115−16xf36λ)4/3, (2.45)

and leads to tachyon growing exponentially at large ,

 τ(r)∼r→∞τ0eCr (2.46)

where is a known constant (see Appendix B) and parametrises the solutions. The second choice is given by

 κ(λ)=1(1+λ)4/3,a(λ)=κ(λ)32L2\scriptsize UV[1+(89b0+1)λ+λ2] (2.47)

and for them the leading divergence is

 τ(r)∼r→∞C√r−r1, (2.48)

where the constant is again known and now parametrises the solutions. To select this solution, it is required that the last term in the square brackets in (2.47) grows faster than .

Finally, let us summarize our choices for acceptable potentials. We always keep fixed to the expression (2.36) and choose , , and as follows:

• Potentials I: We take as in equation (2.39), so that the fixed point exists for all . For and we use the choice of equations (2.45), which lead to exponentially diverging tachyon in the IR.

• Potentials II: We take again from equation (2.39), but use the other choice (2.47) for and . Then the tachyon diverges as in the IR.

• Potentials I: We use now the fermionic potential of equation (2.42), which contains the confinement factor. Thus the extremum exists only within the interval . For and we use the choice of equations (2.45), which lead to exponentially diverging tachyon in the IR.

• Potentials II: We use with the confinement factor, but use the other choice (2.47) for and . Then the fixed point exist only for large , and the tachyon diverges as in the IR.

To fully pin down the potentials, we also need to specify the value of (or ) which is used. We choose four reference values:

• (and constant). This is the lower bound of . Actually, exactly zero is not acceptable because the anomalous dimensions of the quark mass and the chiral condensate do not sum up to zero. This case is nevertheless interesting as it is the limit of acceptable solutions.

• . This is the standard choice studied in [22].

• . For constant , this is the largest possible value, for which as .

• (and ) fixed such that the free energy automatically matches with the standard Stefan-Boltzmann result at high temperature with the correct number of degrees of freedom (see Eq. (2.32) and the discussion in Sec. 3.4 below).

An ongoing work [45] studies the meson spectra in this model. As it turns out, the potentials I and I admit linear “Regge” trajectories, so that the quadratic masses are asymptotically linear in the excitation number, , independently of the other quantum numbers. Potentials II and II, however, have linear trajectories only in the glueball sector, while the other trajectories are quadratic, . As linear trajectories are expected in QCD, this observation favors potentials I and I.

IR fixed point and the BF bound for the tachyon

Now that the potentials are defined, one can check that they satisfy an important requirement: they permit the determination of the bulk dilaton mass and, equating this with the Breitenlohner-Freedman (BF) instability bound, the determination of the start of the conformal window. Take (there is no chiral symmetry breaking in the conformal window) and note that at small , . However, grows faster and the conformal window starts at the value defined by the vanishing derivative

 V′g(λ∗)−xfV′f0(λ∗)=0. (2.49)

Given one defines an IR AdS radius

 12L2\scriptsize IR=Vg(λ∗)−xfVf0(λ∗),L\scriptsize UV>L\scriptsize IR% . (2.50)

The tachyon mass at in units of becomes

 −m2\scriptsize IRL2\scriptsize IR=24a(λ∗)κ(λ∗)[Vg(λ∗)−xfVf0(λ∗)]. (2.51)

Gravity solutions with are stable when ; the conformal window thus starts when (2.51), as a function of , has the value 4.

Eq. (2.51) can be evaluated for the two choices of above. For the choice (2.45) (types I and I) the equation becomes

 36[1+136(115−16xf)λ∗]4/3L2\scriptsize UV[Vg(λ∗)−xfVf0(λ∗)]=4 . (2.52)

For the choice (2.47) (types II and II), the -equation (2.51) has the form

 36[1+127(115−16xf)λ∗+λ∗2]L2\scriptsize UV[Vg(λ∗)−xfVf0(λ∗)]=4 . (2.53)

The values of