1 Introduction

ITP-UU-08/79, SPIN-08/61

The Klebanov-Strassler model with massive

dynamical flavors

Francesco Bigazzi , Aldo L. Cotrone , Angel Paredes ,

Alfonso V. Ramallo

Physique Théorique et Mathématique and International Solvay Institutes, Université Libre de Bruxelles; CP 231, B-1050 Bruxelles, Belgium.

Institute for theoretical physics, K.U. Leuven; Celestijnenlaan 200D, B-3001 Leuven, Belgium.

Institute for Theoretical Physics, Utrecht University; Leuvenlaan 4, 3584 CE Utrecht, The Netherlands.

Departamento de Fisica de Particulas, Universidade de Santiago de Compostela and Instituto Galego de Fisica de Altas Enerxias (IGFAE); E-15782, Santiago de Compostela, Spain.

fbigazzi@ulb.ac.be, Aldo.Cotrone@fys.kuleuven.be, A.ParedesGalan@uu.nl, alfonso@fpaxp1.usc.es

Abstract

We present a fully backreacted D3-D7 supergravity solution dual to the Klebanov-Strassler cascading gauge theory coupled to a large number of massive dynamical flavors in the Veneziano limit. The mass of the flavors can be larger or smaller than the dynamically generated scale. The solution is always regular at the origin of the radial coordinate and as such it can be suitably employed to explore the rich IR physics of the dual gauge theory. In this paper we focus on the static quark-antiquark potential, the screening of chromoelectric charges induced by the dynamical flavors, the flux tube breaking and the mass spectrum of the first mesonic excitations. Moreover, we discuss the occurrence of quantum phase transitions in the connected part of the static quark-antiquark potential. Depending on the ratio of certain parameters, like the flavor mass, with respect to some critical values, we find a discontinuous (first order) or smooth transition from a Coulomb-like to a linear phase. We evaluate the related critical exponents finding that they take classical mean-field values and argue that this is a universal feature of analogous first order transitions occurring in the static potential for planar gauge theories having a dual supergravity description.

1 Introduction

The quarks in QCD can be distinguished between light () and heavy (), depending on their mass being smaller or (much) larger than the scale dynamically generated via dimensional transmutation. The main vacuum polarization effects related to the flavors are due to the three light quarks, while the others, which can be mainly considered as “probes” of the theory, can be neglected in the path integral with a good approximation.

Nowadays, the only systematic, first-principles-based, non-perturbative approach to low energy QCD is provided by the study of extrapolations to the continuum of numerical simulations of the theory on a Euclidean space-time lattice of finite volume [1]. On the lattice it is practically extremely hard to account for the vacuum polarization effects due to the light quarks and so many results are obtained using an approximation where they are treated as probes. This is the so-called “quenched” approximation. There are certainly indications about the possibility of overcoming this limit in the future and some partial results towards “unquenching” the lattice are known (see for example [2]).

In the meantime we have to take in mind that most of what we know from the lattice is about quenched QCD. This theory is strictly confining, like pure Yang-Mills, while the same is not true for real QCD: due to the presence of the dynamical (light) quarks, an external quark-antiquark pair will not experience an indefinitely linear potential at large distances. Instead, since a pair of dynamical flavors can be popped out of the vacuum, the initial state will decay into a pair of heavy-light mesons for distances larger than a certain “screening length” (at which the flux tube becomes sufficiently energetic for the decay to happen).

Remarkably, the string/gauge theory correspondence, which aims to be a complementary approach towards explaining the non-perturbative dynamics of QCD, offers a set of simple tools to analyze particular unquenched gauge theories in certain regimes. This is certainly not an ideal setting at the moment, since the present computational methods allow us to give a string dual description to theories which are at most (supersymmetric) extensions of planar () QCD, where the phenomenologically interesting sector is coupled to spurious matter or has some higher dimensional UV completion. Nevertheless, the string approach can give valuable insights into many features of strongly coupled gauge theories, in regimes inaccessible by other methods.

In this paper we present an exact (and mostly analytic) fully backreacted D3-D7 supergravity solution dual to the 4d Klebanov-Strassler (KS) cascading gauge theory [3], coupled to a large number of dynamical fundamental matter fields which are massive and non-chiral (relevant related studies are in Refs. [4]-[24]). The unflavored theory is known to be confining and to share many notable properties with 4d SYM. The flavored model flows in the IR to a SQCD-like theory. The dual supergravity solution allows one to consider, without limitations, either light () or heavy () dynamical flavors. Remarkably, the solution is always regular at the origin of the radial coordinates and as such it is a very promising tool to explore the IR dynamics of the dual gauge theory.

With this aim, we begin to explore some relevant physical observables, concentrating in particular on their dependence on the flavor parameters. We first focus on the static quark-antiquark potential and study its behavior as a function of the number and the masses of the flavors, keeping fixed different relevant field theory scales. We show how the solution accounts for the string breaking and screening effects due to the dynamical flavors. In particular it enables us to provide a first qualitative study of the behavior of the screening lengths as functions of the flavor parameters. We also start up an analysis of the mesonic spectrum, by considering fluctuations of the worldvolume gauge field of a -symmetric D7-brane probe corresponding to massless flavors. We show how it is possible to study quantitatively how the spectrum varies w.r.t. to the number of sea flavors and their masses. Previous studies along these lines in models with flavor backreaction can be found in [5, 8, 17, 18, 22, 24].

Finally, we analyze the interesting phenomenon of the occurrence of first order quantum phase transitions in the “connected” part111In the dual picture this refers to the macroscopic open string describing the metastable state when the mixing with the final heavy-light mesons , i.e. the “disconnected” string configurations, is artificially turned off. of the static quark-antiquark potential. These transitions are discontinuous changes in the slope of the potential and actually occur in several models with and without flavors when at least two separate physical scales are present [25, 26, 27, 28, 17, 18, 21, 24]. For the case at hand the potential passes discontinuously from a Coulomb-like to a linear behavior. The transition disappears, i.e. the potential becomes smooth, for flavor masses above a certain critical value, or by varying some other mass parameters in the theory. In the present setup it is possible to evaluate the related critical exponents and to show that they take the classical mean-field values. We argue, by means of catastrophe theory, that this is a universal feature of every first order transition discovered, in different (also unflavored) models, in the static potential holographically evaluated by means of string theory.

1.1 Techniques and structure of the paper

The addition of fundamental matter to the KS model is realized, following the general suggestion of [29], by means of space-time filling D7-branes which are holomorphically embedded (so as to preserve supersymmetry) and wrapped on non-compact submanifolds of the transverse space [30]. Just as in lattice gauge theory, the task of adding flavors in the stringy setup becomes computationally simpler if the vacuum polarization effects due to the fundamental matter fields are neglected. The quenched approximation is realized by neglecting the backreaction of the flavor D-branes on the background and treating them as external probes.

In order to go beyond the quenched approximation we use a simple technique which was introduced in [6, 7]: we homogeneously smear the flavor branes along their transverse directions. This operation is sensible only if the number of such branes is very large, as happens in the Veneziano regime (where with fixed). In the smeared setups the flavor symmetry group is generically broken to a product of abelian ones, but this limitation does not spoil many features of physical interest in the related models.

When the flavors are massless the related smeared branes reach the origin of the transverse space. At this point the flavor symmetry is generically enhanced since the flavor branes overlap. In the known cases this is accompanied by the presence of a (good) singularity in the dual string solutions. This kind of singularity can be avoided in the massive case, where the smeared branes generically extend up to a certain finite distance from the origin along the radial direction and there is no special point where the flavor symmetry is fully enhanced. For this reason it is extremely interesting to focus on smeared-flavor-brane setups where the dual gauge theories are coupled to massive dynamical flavors.

Using the smearing technique, the addition of massless dynamical flavors to the KS gauge theory, giving a solution with a (good) singularity in the IR, was considered in [11]. An approximate solution corresponding to the inclusion of massive dynamical flavors to the Chamseddine-Volkov-Maldacena-Nunez confining theory [31], was given in [17] and was regular in the IR by construction. In order to build up an exact solution accounting for massive flavors in the KS case, we have to evaluate the density distribution of the smeared flavor D7-branes. This depends on a function of the radial variable, , which accounts for the effective number of flavor degrees of freedom at a given energy scale. Remarkably, despite the complicated setting, we are able to find the explicit expression of . The precise knowledge of this function enables us to write up the fully backreacted D3-D7 solution. Previous calculations of the analogous function and the derivation of the corresponding backgrounds in the singular conifold case were performed in [18, 22].

This paper is organized as follows. In section 2 we derive the function , the main formulas being (2.14), (2.15), (2.16). In section 3 we calculate the full supergravity solution dual to the KS model with dynamical massive flavors (the solution can be found in section 3.3). In section 4 we study the static quark-antiquark potential and the screening lengths. In section 5 we analyze the first order quantum phase transitions in the static quark-antiquark potential. In section 6 we focus on some mesonic mass spectra. We end up in section 7 with some concluding remarks and a sketch of possible future research lines. The paper includes various appendices where many details of the calculations and validity checks are provided.

2 Massive non-chiral flavors and smeared D7-branes on the deformed conifold

The deformed conifold is a regular six dimensional non compact manifold defined by the equation in . When the complex deformation parameter is turned off, it reduces to the singular conifold, which is invariant under complex rescaling of the , has isometry and topology. The deformation parameter breaks the scale invariance, produces a blown-up at the apex of the conifold and breaks the isometry to .

The low energy dynamics of regular and fractional D3-branes on the deformed conifold is described by a cascading 4d gauge theory with gauge group and bifundamental matter fields transforming as doublets and interacting with a quartic superpotential . The KS solution [3] is relevant for the case, where is an integer. The related theory develops a Seiberg duality cascade which stops after steps when the gauge group is reduced to . The regular KS solution precisely accounts for the physics of an -symmetric point in the baryonic branch of the latter theory, which exhibits confinement and breaking due to the formation of a gluino condensate . The complex parameter is the geometric counterpart of this condensate.

Let us consider the addition of fundamental degrees of freedom to the theory. This can be realized by means of suitably chosen D7-branes. A relevant example is given by D7-branes wrapping the holomorphic 4-cycle defined by an equation of the form

 z1−z2=2^μ. (2.1)

It was shown in [30] that this embedding is -symmetric and hence preserves the four supercharges of the deformed conifold theory.

A D7-brane wrapping the 4-cycle defined above is conjectured to add a massless (if ) or massive (anti) fundamental flavor to a node of the KS model. The resulting gauge theory is “non-chiral” because the flavor mass terms do not break the classical flavor symmetry of the massless theory. The related perturbative superpotential is, just as in the singular conifold case [32],

 W=WKW+^h1~q1(A1B1−A2B2)q1+^h2~q2(B1A1−B2A2)q2+ki(~qiqi)2+m(~qiqi), (2.2)

where we have considered a -invariant setup with two stacks of D7-branes adding the same number of fundamental degrees of freedom (with same masses) to both nodes. Thus the complex mass parameter in is mapped to the geometrical parameter .

In the singular conifold case the massive embedding explicitly breaks the scale invariance and the isometry of the background geometry. This is related to the explicit breaking of conformal invariance and symmetry due to the mass terms in the dual gauge theory. The embedding equation also breaks part of the non abelian symmetry group of the conifold to a diagonal subgroup.

2.1 The D7-brane profile

The metric of the deformed conifold is usually written as

 ds26=12ϵ4/3K(τ)[13K3(τ)(dτ2+(g5)2)+cosh2(τ2)((g3)2+(g4)2)+sinh2(τ2)((g1)2+(g2)2)], (2.3)

where

 K(τ)=(sinh(2τ)−2τ)1/321/3sinh(τ), (2.4)

and

 g1 = −sinθ1dφ1−cosψsinθ2dφ2+sinψdθ2√2, g2 = dθ1−sinψsinθ2dφ2−cosψdθ2√2, g3 = −sinθ1dφ1+cosψsinθ2dφ2−sinψdθ2√2, g4 = dθ1+sinψsinθ2dφ2+cosψdθ2√2, g5 = dψ+cosθ1dφ1+cosθ2dφ2. (2.5)

The range of the angles is , , , while . For the metric asymptotes the singular conifold one. In terms of these coordinates the non-chiral embedding can be written as

 Θ1sinhτ2−iΘ2coshτ2=^μϵ, (2.6)

where

 Θ1 = sinθ12sinθ22cosφ1+φ2−ψ2−cosθ12cosθ22cosφ1+φ2+ψ2, Θ2 = sinθ12sinθ22sinφ1+φ2−ψ2+cosθ12cosθ22sinφ1+φ2+ψ2, (2.7)

and . From these equations it follows that the profile of the D7-branes has a non trivial radial dependence: the branes extend all along the radial direction up to a minimum distance which depends on the relative phase of the and parameters.

If is purely imaginary (resp. real), the embedding equations imply (resp. ) and (resp. ). For generic phases . Notice that in the “completely misaligned” case where is purely imaginary, - i.e. the D7-brane reaches the tip of the deformed conifold - if .

Let us now define as the absolute minimal value of . Related to this, one can introduce a parameter , the absolute minimum flavor “constituent mass”,222This is how the parameter is usually called in the literature. In the present context we adopt the same name with an abuse of language. defined as the energy of an hypothetical straight string stretched along the radial direction from to . If , then . As discussed above, the dimensional parameters and can be related to the bare flavor mass (see eq. (2.2)) and the fundamental scale of the dual gauge theory. Thus the relation can be interpreted as . Though at the level of the “constituent mass” we do not see differences between the and the cases, we will see that the non zero bare mass parameter influences the density distribution of the flavor branes also when . This will thus mark a difference with the setup.

2.2 The density distribution for smeared D7-branes

Acting with an rotation on the embedding equation , we obtain the generalized embedding equation

 ¯pz1−pz2+¯qz3+qz4=2^μ≡2|^μ|eiβ, (2.8)

where span a unit 3-sphere

 p=cosθ2ei(χ+ϕ2),q=sinθ2ei(χ−ϕ2), (2.9)

and , , , .

Let us now consider a maximal symmetric smeared distribution of D7-branes generally embedded as above. By “maximal” we mean that the distribution will not only be invariant under the isometry of the deformed conifold, but also under the symmetry , under shifts of the angle, which is broken (to a subgroup) by the deformed conifold geometry. We will thus homogeneously distribute the D7-branes along as well as along the phase of the mass term . We will instead take the modulus (hence the modulus of the flavor mass parameter in the dual field theory) to be fixed. Smearing along will cause different D7-branes to reach different minimal distances from the origin. The whole distribution will end up at the absolute minimal distance .

The density distribution of the smeared D7-branes is given by

 Ω=∫σθ,ϕ,χ,β(δ(f1)δ(f2)df1∧df2)dθdϕdχdβ, (2.10)

where we have introduced the properly normalized density function and , are the two real constraints implied by the complex equation (2.8) (see appendix A).

The symmetries strongly constrain the form of . As was shown in [9], the only possibility for an exact two-form preserving is, in the present setup,

 Ω=Nf(τ)4π(sinθ1dθ1∧dφ1+sinθ2dθ2∧dφ2)−˙Nf(τ)4πdτ∧(dψ+cosθ1dφ1+cosθ2dφ2), (2.11)

where the the dot means derivative w.r.t. . The function counts the effective number of dynamical flavors at a given energy scale, holographically related to . It crucially depends on the particular kind of smeared embedding. Referring to appendix A for the details of the non trivial and very instructive calculation and defining

 x≡coshτ,μ≡|^μ||ϵ|, (2.12)

we find that in the present setup the function is the solution of the first order equation

 (2.13)

where is the Heaviside step function and

 I1 = 4π(1+x2), I2 = x√(1+2μ2−x)(1−2μ2+x)(x2−1)+2μ2(1+x2)arctan[√(1+x)(1+2μ2−x)(x−1)(1−2μ2+x)].

In the massless case we have [11]. Moreover, for we have .

Let us now split (2.13) into two regions. In region I, (i.e. ), we find the following simple solution

 N(I)f(x)=Nf[1−2μ2xx2−1], (2.14)

where the integration constant is fixed by consistency so that . Notice that in the large limit (with ), this function asymptotes to the expression found in [22], for the flavored version of the singular conifold Klebanov-Witten (KW) model [33].

In region II, (i.e. ), we have a complicated expression in terms of Elliptic integrals of the first and third kind ( and respectively)

 (2.15)

where

 A1(x,μ2) = −14μ2√(x+1−2μ2)(2μ2+1−x)(x+1)(x−1)−xx2−1arctan[√(1+x)(1+2μ2−x)(x−1)(x+1−2μ2)], A2(x,μ2) = −i2μ4(μ2−1)F[arcsin(√μ2(1+x)(1+μ2)(x−1))|μ4−1μ4]+ (2.16) −iμ4Π[μ2+1μ2;arcsin(√μ2(1+x)(1+μ2)(x−1))|μ4−1μ4].

In (2.15) we have fixed the integration constant by imposing continuity at , i.e. . This condition is satisfied since .

If and so , we also have that is vanishing for . Let us check continuity at . First, notice that

 A1(2μ2−1,μ2)=−π8(2μ2−1)μ2(μ2−1), (2.17)

so that the above mentioned continuity condition amounts to having

 1−4μ2π[A2(2μ2−1,μ2)−A2(2μ2+1,μ2)]=0. (2.18)

We have checked numerically that this condition is indeed satisfied and, more precisely, that

 1−4μ2π[A2(2μ2−1,μ2)−A2(2μ2+1,μ2)]=−2Θ[1−μ], (2.19)

which thus vanishes for .

Relevant plots of can be found in figure 1.

For the shape of resembles that of a “smoothed-out” Heaviside step function . In fact, as we have anticipated, the function counts the effective number of flavor degrees of freedom at a given energy scale. At energies smaller than the flavor mass, the fundamental fields can be integrated out and the theory resembles the unflavored one. At higher energies the masses can be neglected and the theory looks like the massless-flavored one (for which ). In other contexts where the density distribution of the flavor branes is difficult to evaluate, the Heaviside step function can be fruitfully used to construct approximate solutions. This is what was done in [17] for the massive-flavored CVMN solution.

Let us stress that also when and so , , there is a non trivial density distribution of branes. This marks a difference with the case studied in [11]. Of course approximating with an Heaviside step function in this massless case would simply replace our solution with the massless one for every .

3 The backreacted KS solution with massive flavors

In this section we present the new supergravity solutions accounting for the full backreaction of fractional and regular color D3-branes, as well as of smeared flavor D7-branes on the deformed conifold. The solutions follow from an action which is the sum of the bulk type IIB supergravity and the flavor brane actions. Following a suggestion in [34] the action for the D7-branes is just taken as the sum of the Dirac-Born-Infeld (DBI) and Wess-Zumino (WZ) terms. This is actually an approximation which is sensible only if the effective coupling is small. This is the case in localized setups if or in the smeared setups (where can be of the same order of ) due to the effective suppression of the coupling by the large transverse volume [18, 19].

3.1 The ansatz

In order to present the ansatz for the full background, let us first introduce the one-forms and () as follows

 σ1=dθ1,σ2=sinθ1dφ1,σ3=cosθ1dφ1, ω1=sinψsinθ2dφ2+cosψdθ2,ω2=−cosψsinθ2dφ2+sinψdθ2, ω3=dψ+cosθ2dφ2. (3.1)

Here the angles are the same as in the deformed conifold.

The Einstein frame metric ansatz has the same warped form as in the massless case [11]

 ds2 = h−1/2(τ)dxμdxμ+h1/2(τ)ds26, ds26 = 19e2G3(τ)(dτ2+g25)+e2G2(τ)(1−g(τ))(g21+g22)+e2G2(τ)(1+g(τ))(g23+g24),

where denotes the four-dimensional Minkowski metric and (), and are five unknown radial functions. Quite nicely, the embedding equation expressed in terms the “deformed conifold variable” looks the same in terms of the “backreacted ansatz variable”. See appendix B for details.

As for the dilaton and the forms we will adopt the same ansatz as in [11], modulo the substitution of with the function evaluated in the previous section. In units we have

 F5 = dh−1(τ)∧dx0∧⋯∧dx3+Hodgedual,ϕ=ϕ(τ), B2 = α′M2[fg1∧g2+kg3∧g4], H3 = α′M2[dτ∧(˙fg1∧g2+˙kg3∧g4)+12(k−f)g5∧(g1∧g3+g2∧g4)], F1 = Nf(τ)4πg5, F3 = α′M2{g5∧[(F+Nf(τ)4πf)g1∧g2+(1−F+Nf(τ)4πk)g3∧g4]+ (3.2)

where is the fractional D3-brane Page charge and , , are functions of the radial coordinate (and where the dot denotes derivative with respect to ).

Notice that, consistently, , where is the D7-brane density distribution form given in eq. (2.11). This and the other modified Bianchi identities

 dF3=H3∧F1−Ω∧B2, dF5=H3∧F3−12Ω∧B2∧B2, (3.3)

follow from the WZ term of the smeared D7-brane action (see appendix C).

3.2 The BPS equations

The modified Bianchi identity for in (3.3) is automatically satisfied by the ansatz, while that for reduces to a first order differential equation for the warp factor

 ˙he2G1+2G2=−α′24M2[f−(f−k)F+Nf(τ)4πfk]+N0, (3.4)

where is an integration constant that we will set to zero as in [3] and [11]. The previous equation is the same as the one obtained in [11] with the substitution . In general, the BPS equations following from the bulk fermionic supersymmetric variations and from the Bianchi identities are of exactly the same form as those in [11], with the only substitution of with the function . This is also due to the fact that, despite the modified Bianchi identities of the forms in the massive setup differing from those in the massless case, the fermionic supersymmetric variations only contain the and not the .

In this way one arrives at the same algebraic constraint for as in [11]: . Its two solutions and correspond to the singular (and resolved) conifold and to the deformed conifold respectively. Here we focus only on this latter case since we want to have a regular solution at .

The BPS equations for the 6d metric functions are

 ˙G1 = 118e2G3−G1−G2+12eG2−G1−12eG1−G2, ˙G2 = 118e2G3−G1−G2−12eG2−G1+12eG1−G2, ˙G3 = −19e2G3−G1−G2+eG2−G1−Nf(τ)8πeϕ, (3.5)

while for the dilaton we have

 ˙ϕ=Nf(τ)4πeϕ. (3.6)

Notice that, just as in the massless case, by defining we get the simple equation , from which, up to an integration constant (that we fix to zero as in the massless case) it follows that

 eG1−G2=tanhτ,→g−1=coshτ. (3.7)

Taking this result into account, for the flux functions we have

 ˙k = eϕ(F+Nf(τ)4πf)coth2τ2, ˙f = eϕ(1−F+Nf(τ)4πk)tanh2τ2, ˙F = 12e−ϕ(k−f), (3.8)

supported by the algebraic constraint

 (3.9)

In order to solve the above set of equations we will have to distinguish between the two possible cases: 1) , i.e. and , with the running function being equal to zero for ; 2) , i.e. and with the running function being non trivial up to .

3.3 The solution

Let us start by considering the (i.e. ) case. Correspondingly, there is a region where the effective D7-brane charge is zero. In that region, requiring regularity, the solution is just (a slight generalization of) the unflavored KS one. Since , the dilaton does not run (see eq. (3.6)) and the flux functions are just the KS ones, modulo an overall constant

 eϕ = eϕIR=constant,F=sinhτ−τ2sinhτ, f = eϕIRτcothτ−12sinhτ(coshτ−1)≡eϕIRfKS, k = eϕIRτcothτ−12sinhτ(coshτ+1)≡eϕIRkKS. (3.10)

The metric is a warped product of 4d Minkowski and the deformed conifold with deformation parameter (2.3), since

 e2G1=e2G2tanh2τ,e2G2(τ)=coshτ4ϵ4/3IRK(τ),e2G3(τ)=32ϵ4/3IRK2(τ). (3.11)

The warp factor is given by

 h(τ)=223α′2M2ϵ83IR⎡⎢⎣h0−eϕIR∫τ0(ξcothξ−1)(sinh2ξ−2ξ)13sinh2ξdξ⎤⎥⎦, (3.12)

where is an integration constant. In [3], was fixed by imposing . Since the solution we are considering is only valid up to we cannot fix the integration constant in the same way.

The above solution has to be continuously glued to the one obtained in the region where the effective D7-brane charge is non zero. The function has a very non trivial expression in general, so we will have to perform some numerical integration. For the dilaton, for example, from eq. (3.6), it follows that

 e−ϕ(τ)=14π∫τ0τNf(ξ)dξ, (3.13)

where the is a point where blows up. A simple analytic expression can be obtained in the region where is given by eq. (2.14)

 eϕ(τ)=4πNf1(τ0−τ)+2μ2(1/sinh(τ0)−1/sinh(τ)),(τ>arccosh(2μ2+1)). (3.14)

Just as in the massless-flavored KS [11] or in the flavored KW cases [9, 18, 22], where was related to a Landau pole in the dual gauge theories, our solution cannot be continued up to infinity: .

Requiring continuity at we find

 e−ϕIR=14π∫τ0τqNf(ξ)dξ, (3.15)

which explicitly depends on and .

A remarkable feature of the present setup is that also in the effectively flavored region most of the solution can be given in an analytic way. In fact after some algebra we find the following results for the metric functions

 e2G1=ϵ43UV4e−ϕ3sinh2τcoshτK(τ),e2G2=ϵ43UV4e−ϕ3coshτK(τ),e2G3=32ϵ43UVe−ϕ3K(τ)2, (3.16)

where

 K(τ)≡[sinh2τ−2τ+η(τ)]13213sinhτ, (3.17)

and . The function is thus a constant in the unflavored region. By requiring continuity at we fix this constant to zero so that

 η(τ)=eϕ4π∫ττq(sinh2ξ−2ξ)Nf(ξ)dξ, (3.18)

and

 ϵUV=ϵIReϕIR/4,K(τq)=K(τq). (3.19)

The metric in the region is thus a warped product of Minkowski 4d and a slight deformation (driven by ) of the deformed conifold metric

 ds26UV=12ϵ43UVe−ϕ(τ)3K(τ)[13K3(τ)(dτ2+(g5)2)+cosh2(τ2)((g3)2+(g4)2)+ +sinh2(τ2)((g1)2+(g2)2)]. (3.20)

In appendix D we will verify that this metric reduces to the one found in [11] when the flavors are massless. The warp factor is given by

 h(τ)=−4M2α′2ϵ8/3UV∫ττqe23ϕ(ξ)sinh2ξK(ξ)[f(ξ)−(f(ξ)−k(ξ))F(ξ)+Nf(ξ)4πf(ξ)k(ξ)]dξ+h1, (3.21)

where the integration constant is constrained by the continuity condition at

 h1=223α′2M2ϵ83IR⎡⎢⎣h0−eϕIR∫τq0(ξcothξ−1)(sinh2ξ−2ξ)13sinh2ξdξ⎤⎥⎦. (3.22)

There is something important to notice here: for large values of , diverges as . Thus, (this happens also in the massless-flavored KS case). Since the metric is only well defined for , we conclude that there exist some maximal value of the radial coordinate where vanishes and a singularity appears. This behavior at could be connected to the presence of a duality wall [11].

For the flux functions things are simpler. By using the constraint equation (3.9) and imposing continuity at , we promptly get

 f = eϕτcothτ−12sinhτ(coshτ−1), k = eϕτcothτ−12sinhτ(coshτ+1), F = sinhτ−τ2sinhτ. (3.23)

If the D7-branes reach the tip of the cone, i.e. if (i.e ), there is no effectively unflavored region, and thus the solution has the non trivial dependence in the whole region. The 6d part of the warped metric is again a generalized () deformed conifold with parameter . In principle in this case we would not need a condition analogous to the first equation in (3.19). In practice, by continuity with the solutions we will rescale as in (3.19).

Sample plots of the relevant functions in the solution are given in figure 2.

In order to completely verify that these results are correct one has to check that the BPS equations solve the equations of motion of gravity, dilaton and forms. To this aim a general result in [35] helps: it was shown in that paper that, for general backreacted solutions with metric having a warped form, the solutions of the first order equations following from the bulk fermionic supersymmetry variations and from the (source modified) Bianchi identities, are also solutions of the Einstein equations and of the second order equations of motion for the dilaton and the form fields, provided the sources (smeared or localized) are supersymmetric (i.e. satisfy preserving -symmetry conditions). Our setup indeed satisfies these conditions. In appendix E we give the relevant ingredients to perform the consistency check explicitly. See also [14] for relevant related comments.

3.4 Validity and regularity of the solution

We can now study the Ricci scalar and the square of the Ricci tensor in the string frame, to check the validity and regularity of the solutions we have found above. As a first thing, one can check that since (where for we used the rescaling in (3.19)), we have that . Thus, as usual in D3-D7 systems, the supergravity solution is reliable in the regime . The same observation applies also to the massless-flavored KS solution.

Both curvature invariants diverge at , though they stay “small” up to values of very close to (for example up to if we fix ).333Note added: there is actually another particular point (but very close to ) where the curvature invariants are finite but the holographic -function has a singularity. One can argue the need to place a UV cutoff, smaller than , when computing observables in this background [36]. In order to study the behavior near we need to know the behaviors of the functions in the case (for the IR solution is the KS one, so the the solution is certainly regular at ). Clearly everything depends on the behavior of , which goes as