1 Introduction

ITP-UU-09/12, SPIN-09/12

KUL-TF-09/11

Screening effects on meson masses

from holography

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 Física de Partículas, Universidade de Santiago de Compostela and Instituto Galego de Física de Altas Enerxías (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 study the spectra of scalar and vector mesons in four dimensional strongly coupled SQCD-like theories in the Veneziano limit. The gauge theories describe the low energy dynamics of intersecting D3 and D7-branes on the singular and deformed conifold and their strong coupling regime can be explored by means of dual fully backreacted supergravity backgrounds. The mesons we focus on are dual to fluctuations of the worldvolume gauge field on a probe D7-brane in these backgrounds. As we will comment in detail, the general occurrence of various UV pathologies in the D3-D7 set-ups under study, forces us to adapt the standard holographic recipes to theories with intrinsic cutoffs. Just as for QED, the low energy spectra for mesonic-like bound states will be consistent and largely independent of the UV cutoffs. We will study in detail how these spectra vary with the number of the fundamental sea flavors and their mass.

1 Introduction

Holography is nowadays a standard and powerful method to investigate properties of some strongly coupled gauge theories. Most of the studies of flavor physics in this context have focused on the quenched approximation where the internal quark loops are neglected. The extension of these investigations to the unquenched cases, where the full dynamics of the flavors is included, is of obvious interest. The main focus of this paper is the study of mass spectra of low spin () mesons in certain strongly coupled SQCD-like theories in the unquenched Veneziano regime, where the number of colors and the number of flavors are both taken to be very large, with their ratio taken to be fixed. The aim is to extract the dependence of the spectra on the number, , of sea flavors and on their mass, .

The SQCD-like models we will consider describe the low energy dynamics at the 4d intersection of “color” D3-branes and homogeneously smeared “flavor” D7-branes on the singular and on the deformed conifold. The models are the flavored unquenched versions of the conformal Klebanov-Witten (KW) [1] and the confining Klebanov-Strassler (KS) [2] ones. The strong ’t Hooft coupling regime of these theories is mapped to some dual backgrounds, arising as the full backreaction of the color and the flavor D-branes, with D7-brane sources. The supergravity background dual to the KW (resp. KS) model coupled to chiral (resp. non chiral) massless dynamical flavors was found in [3] (resp. [4]). The solution with chiral massless flavors in the KS case, which we will not consider, appears in [5]. These massless-flavored solutions have generically (good) singularities at the origin of the transverse radial coordinate. The singularity is avoided in the massive case. The supergravity dual of the KW model coupled to chiral (resp. non chiral) dynamical massive flavors was found in [6] (resp. [7]). The relevant solution in the KS case with non chiral massive flavors was found in [8].

In order to extract the mesonic spectra we will probe these backgrounds with an external D7-brane and study the fluctuations of a selected set of decoupled modes (dual to scalar and vector mesons) of the gauge field on its worldvolume. This study will require a careful treatment of the boundary conditions of the fluctuating fields, in connection with the fact that the backreacted D3-D7 backgrounds - unlike the corresponding unflavored ones - have various UV pathologies. In particular, we will show that there is a singularity in the holographic -function at finite radial position. Just as in QED, which has a UV Landau pole, the spectra of the various bound states will nevertheless be meaningful and independent of the cutoffs (up to corrections suppressed by the UV scale).

Since the goal of this paper is to study the dependence of the mesonic spectra on the number of sea flavors and their masses, we will have to decide how to interpret our results, i.e. how to compare different theories with different flavor parameters. As already noticed in [8], there is no obvious natural scale or coupling which can be assumed to stay fixed when varying those parameters. In any case, the comparison between two theories will necessarily require us to make a choice of what to keep fixed. The comparison will strongly depend on this choice. To disentangle the possible different conclusions we will need some explicit formula for the mesonic masses as a function of the physical parameters in the theory.

We will parameterize the meson masses as where is a dimensionful scale which can be “measured” in the IR and which we will take fixed when comparing different theories. This scale will be related to the “coupling” and the effective “string tension” at the probe quark mass scale. The coefficients will then be determined from a numerical analysis.

Our results show that the ’s receive small “radiative” corrections due to dynamical flavor loops and decrease as the effective coupling (where ) is increased (or if the mass of the sea flavors is decreased). These small “quantum effect” corrections resemble the hydrogen atom Lamb shift in QED. In section 5 we will argue that, for the flavored KW models, these corrections could be related to the small variation of the effective number of (adjoint plus bifundamental) degrees of freedom due to the internal flavor loops. In the KS cases they could be related to corrections to the glueball (or Kaluza Klein) mass scale. The decrease of the ’s depends on the choice of keeping fixed the coupling at the probe quark mass: had we kept fixed the coupling at some larger UV scale, larger could have yielded larger meson masses.

The structure of the paper is the following. We will start, in section 2, by examining the general UV behavior of the D3-D7 models under study. We will show how non trivial UV cutoffs emerge when the holographic -function is considered. Moreover, we will outline the general limits of our analysis. We will then focus on scalar and vector mesons in the flavored conifold models. In section 3, combining the numerical shooting technique and the WKB analysis, we will extract the spectra of mesons having massive constituent flavor fields in the (flavored) KW models. Having chosen a prescription to compare different theories we will study how the sea flavors (either massless or massive) affect the mesonic spectra. In section 4 we will present the results of analogous studies in the flavored KS models, limiting our analysis to the case of mesons with massless fundamental constituents. In section 5 we will discuss a possible way to interpret our results. We will end in section 6 with a set of concluding remarks. Various technical details and some comments on high spin mesons will be left to the appendices.

2 General comments

A common feature of D3-D7 set-ups, both in the case of smeared and localized [9] D7-branes, is a running dilaton in the dual supergravity solutions. The dilaton, and consequently the effective string coupling, typically increases with the transverse radial variable of the backgrounds and blows up at a certain point . The solutions can thus be defined at best only in a region where the dilaton can stay small.111Just to fix the notation consistently with the following sections, we will call (resp ) the radial variable in the flavored KW (resp. KS) solutions and we will set (resp. ). In our conventions large values of are mapped to large energy scales in the dual field theory.

The point can be formally mapped to a UV Landau-pole in the dual field theories. The simplest way to realize it is to consider a localized D3-D7 set-up in flat space. The theory on D3-branes is just SYM, which has exactly zero beta function. We can add flavors to the theory by means of a stack of D7-branes, sharing with the D3-branes the 4d Minkowski directions and extended along a non compact 4-dimensional submanifold of the transverse space. This breaks supersymmetry (to ) as well as conformal invariance. The perturbatively exact coefficient of the beta function for the inverse squared gauge coupling is , which means that the theory has a UV Landau pole. This pathological UV behavior is inherited by D3-D7 set-ups both in orbifold and conifold models.

Apart from the dilaton divergence, the theories we are going to consider have other less explicit UV singularities. In the flavored KW cases the integration constants can be fixed so that the warp factor vanishes exactly at the Landau pole . In the flavored KS cases this is not a consistent choice and the warp factor will diverge badly (going to minus infinity) at . This means that there will be a point where the warp factor vanishes and the metric becomes singular. A possible interpretation of this point is in terms of a duality wall in the dual theory [4]. Since for the warp factor becomes negative, replaces as a sensible UV “end” of the flavored KS backgrounds.

Both the flavored KW and KS models have an even more “hidden” potential UV pathology. It shows up by considering their 5d reduction. Starting with a 10d string frame metric of the form

(2.1)

the reduction on the internal five dimensional manifold (with volume ), gives a 5d Einstein frame metric of the form

(2.2)

where

(2.3)

In standard set-ups, the function , which can be roughly identified with the dual field theory energy scale, monotonically varies with the radial coordinate. This is also required in order for the “holographic -function” [10]

(2.4)

to be finite.222The monotonicity of also plays a crucial role in holographic computations of the entanglement entropy, see [11]. The notations of that paper are used in the equations above.

In both the flavored KW and KS cases, instead, the function is not monotonic: it increases with from zero up to a maximum at a point and then it decreases back to zero (at in the KW cases and at , with in the KS cases). A representative plot is given in figure 1.

Figure 1: The function in the massless-flavored KW model.

This behavior implies that the holographic -function as defined in (2.4) is singular and discontinuous at . As we will show in the following, the -function will be positive and increasing up to where it will asymptotically diverge. At will go to minus infinity and then it will increase to zero (at in the KW case and at in the KS one) for increasing . In order to avoid the apparently pathological region , we will cutoff our D3-D7 backgrounds at .

The occurrence of UV pathologies in the D3-D7 models we are going to consider, does not subtract interest to the study of their IR dynamics. QED, the most notable quantum field theory model with a UV Landau pole, certainly provides excellent and meaningful predictions on the low energy (e.g. atomic) spectra, which do not sensitively depend on the UV cutoff. In order to study, say, the spectrum of bound states of (s)quarks pairs in our D3-D7 conifold models we will just have to reconsider the standard holographic recipes and adapt them to the case where the dual supergravity backgrounds do not have a boundary at infinity. Having discussed the presence of UV pathologies, we stress that the aim of this paper is not to study how to cure them or how to consistently UV complete the theory. Conversely, we focus in computing IR quantities that are only mildly affected by whatever the UV physics is, in a sense that we will make precise below.

In the following we will focus on vector and scalar mesons dual to fluctuations of the gauge field on the worldvolume of a D7-brane probe. The corresponding mass spectrum will be quantized after imposing that the fluctuations be regular at (the minimal radial distance reached by the brane) and vanish at the cutoff . The pathological UV region will thus be excluded and treated as producing an infinite wall in the effective quantum mechanical description of the fluctuating modes. The choice of the cutoff (but not of its maximal value) is nevertheless arbitrary in the range and the consistency of our results will be guaranteed provided we show that the dependence on the cutoff is highly suppressed. Note that we can think about our models as the IR regions of UV consistent theories, providing some UV completions to the backgrounds at hand. This perspective has been employed recently in [12].

Let us conclude with the following remarks. The theories to which we add flavors correspond to the low energy dynamics of regular (and fractional) D3-branes on the (deformed) conifold. They are (cascading) 4d gauge theories with gauge group () and bifundamental matter fields transforming as doublets and interacting with a quartic superpotential . The perturbative superpotential in the flavored case is:

(2.5)

where and are the flavor multiplets and and are the couplings.

The SQCD-like models under study are assumed to be in the Veneziano regime. This means that we take (where is the number of colors) with and fixed. Moreover, the dual supergravity solutions (with DBI+WZ source terms for the smeared D7-branes) will only be reliable provided (as usual) and .333It is worth pointing out that the condition, necessary in all the D3-D7 models, is not mandatory in different set-ups with other kinds of brane intersections. Some examples where can be kept of order one have been studied in [13, 14, 15]. These limits will allow us to consider cases where the effective coupling , weighting the vacuum polarization effects due to the dynamical flavors, is of order one.444Note that the open string coupling on the flavor branes is not : the backreacting branes are smeared in the transverse space, so only a small fraction of them is within a distance of order and the coupling is parametrically smaller than [6, 15].

3 Meson excitations in the KW models with flavor

In this section we will discuss (part of) the spectrum of meson masses in the particular framework of the so-called Klebanov-Witten conformal model [1] and its generalizations with massless [3] and massive [7] unquenched flavors. We start by writing general expressions for the background and excitations and then specialize the study for the different cases. We will close the discussion of the flavored KW models by analyzing the holographic -function in section 3.6.

3.1 The background solution and the excitation equations

We consider solutions of type IIB supergravity coupled to a homogeneously smeared set of D7-branes (we refer the reader to [3, 6, 7] for further details). The ansatz for the fields which take non-trivial values in the solution is, in Einstein frame:

(3.1)

Notice that the 5-form is self-dual and, using the relations:

(3.2)

one can check that it satisfies the quantization condition , where is the internal manifold. If the two-form is the density distribution of the smeared D7-branes (), the action of the gravity+branes system reads [3]:

(3.3)

Then (3.1) provides a supersymmetric solution if:

(3.4)

These equations are valid for the case without unquenched flavors [1] ( such that ), the case with massless unquenched flavors [3] () and the case with massive unquenched flavors [3, 6, 7] (where becomes a suitable -dependent expression depending on the kind of flavor branes under consideration).

In the following, we analyze excitations of a brane probe which preserves the same supersymmetry as the background. Thus, there are two kinds of flavor branes in the generic set-up: the first kind corresponds to dynamical quarks, i.e. the branes backreacting on the geometry, accordingly corresponding to unquenched flavors (in section 3.4 we will denote the value of at their tip). The second kind is a single oscillating probe brane (with tip at ), associated to the quarks which actually constitute the mesons under consideration. Of course it is possible to take this single fluctuating brane to be one out of the backreacting ones but in the following we deal with the more generic case of allowing non-equal masses for dynamical and test quarks (). Since the flavor branes affect the values of the metric, dilaton and p-forms in the solutions, one may wonder whether it makes sense to consider the oscillation of a brane in a background where the closed string fields are taken to be constant. That is indeed the case: a meson is associated to the oscillation of one (or a pair of) flavor brane and not to a collective oscillation of a set of order flavor branes. Thus, the possible backreaction of the mesonic oscillation on the closed string background strictly vanishes in the Veneziano limit.

Let us consider a flavor brane probe of the type first discussed in [16] and corresponding to the embedding ,555An alternative embedding has been considered in [17]. where are two of the complex coordinates defining the conifold. Details and notations are spelled out in appendix A.1. We will only study a reduced subset of all possible mesonic modes. Concretely, we will not discuss fluctuations of the embedding and only the following fluctuations of the worldvolume gauge field, which correspond to a vector and a scalar in the dual gauge theory:

(3.5)

Here, is a constant transverse vector and is the (angular) left-invariant one-form defined in (A.9). We have chosen this particular set of fluctuations for the sake of simplicity, since, as we will see below, it gives rise to relatively simple decoupled differential equations both in the present set-up and in the Klebanov-Strassler case to be discussed in section 4. Also for simplicity, we have not included any angular dependence for the fluctuating gauge field (a truncation which is non-trivially consistent). Even if we restrict ourselves to this very limited subset of all the possible excitations, we expect that the results can point out the general trends on how the presence of dynamical flavors affects the meson masses.

The procedure to obtain the second order equations associated to these modes is outlined in appendix A.2. One gets:

(3.6)
(3.7)

where are the masses for the vector and scalar mesonic excitations under consideration. As it is customary, enforcing appropriate IR and UV behaviors for the functions will select a discrete spectrum for .

Before turning to the study of this issue in several cases, let us introduce a useful parameterization of the meson masses, factoring out a dimensionful scale. Define:

(3.8)

where is the tension of a hypothetical fundamental string stretched at constant and is the ‘t Hooft coupling 666 The value of the tension of eq. (3.9) can be obtained by studying the short distance behavior of the potential, obtained by analyzing a hanging open string in the background metric, see appendix D. Notice that is simply the fundamental string tension redshifted by the string frame warp factor. For the identification with the ‘t Hooft coupling, we have used the orbifold relation . Let us consider , and thus define the ’t Hooft coupling as . One should keep in mind that this orbifold relation is not guaranteed to hold in general. The relations (3.9) should be taken as the definition of , . at the same scale :

(3.9)

The important point is that these are quantities “measured” in the IR, i.e. at the tip of the brane , which is related to the quark mass. The will be towers of numbers which, as we will see, depend on .

Notice, however, that, when writing (3.8), we could have chosen to factor out a different IR dimensionful scale. Basically, this is related to the observation in [18] of the possibility of different definitions of the radius-energy relation in non-conformal theories. For instance, we could have factored out the constituent quark mass to be discussed in section 3.5. Thus, one should keep in mind that when we compute how varies with , we will be computing how meson masses change while keeping fixed.

3.2 Mesons in the quenched approximation

We want to analyze the equations (3.6), (3.7) when the background is just the unflavored KW solution, i.e.:

(3.10)

It is useful to define:

(3.11)

The equations (3.6), (3.7) then read:

(3.12)
(3.13)

where we have used the definitions (3.8) and (3.9). In this conformal case, the actual quark bare mass coincides with the constituent mass defined as the energy of a string stretched from the bottom of the geometry up to the brane tip :

(3.14)

This can be used to rephrase (3.8) as:

(3.15)

Comparing (3.8) to (3.15), we see that .

We can now numerically analyze equations (3.12), (3.13). In each case, requiring regularity at and normalizability in the UV selects a discrete set of values for the ’s. A standard computation using the shooting technique yields:

(3.16)

3.2.1 The Schrödinger potential formalism and WKB estimates

As shown in appendix B, after performing a convenient change of variables, the fluctuations equations (3.12) and (3.13) can be written as Schrödinger equations for some particular potentials. We can then apply the WKB approximation to get an estimate of the mass levels. In the case of the fluctuations described by eqs. (3.12) and (3.13) this analysis leads to the following estimates of :

(3.17)

where and are given by:

(3.18)

In fact, from WKB we only get terms of order and as . Thus, the term inside the square root of the expression for does not come from WKB and we obtained it by fitting the numerical data. In figure 2, we compare these expressions to the first few numerically found eigenvalues.

Figure 2: A plot of the first few values of and obtained numerically (dots) and the approximation given in eq (3.17) (solid line).

3.3 Mesons in the presence of unquenched massless flavors

In this section, the background defined by the equations in (3.4) with constant [3] will be considered. The relevant solution is:

(3.19)

where the represents the incomplete gamma-function.

It is worth commenting on the integration constants appearing in (3.19). As compared to [3], we have explicitly kept the integration constant which fixes the position of the Landau pole (where the dilaton diverges). We have also kept the constant which can be reabsorbed by rescaling the Minkowski coordinates and thus just rescales what one defines as energy. On the other hand, we have set the constant (see eqs. (2.37), (2.38) of [3]) to zero for the sake of IR regularity. Even if the unquenched solution with massless flavors is always IR singular, it was shown in [3] that produces the less severe IR singularity. Most importantly, it was explicitly shown in [6, 7] that by introducing any non-zero mass for the dynamical quarks, a regular IR can be obtained. Taking the massless limit of this family of massive regular solutions yields the condition. Finally, we have set , such that metric and dilaton are both singular at the same point . This UV prescription is not important for the computation of the meson masses: enforcing, instead, for some would just add an additive constant to . As long as , this would only modify the values of the meson masses by quantities exponentially suppressed as . Extended comments regarding the effects of UV prescriptions on the computation of the masses can be found in sections 3.3.1 and 3.3.2.

By inserting (3.19) into (3.6), (3.7), and using again the definitions (3.8), (3.9), one finds the appropriate second order equations. It is easy to check that the resulting spectrum of ’s only depends on , which can be related to the physical quantity by using the explicit solution for the dilaton in (3.19):

(3.20)

The quenched case is recovered when , or .

3.3.1 Schrödinger potentials and UV cutoffs

Our goal now is to study equations (3.6), (3.7) in this background. In each equation we have to demand regularity at for the function that defines the fluctuation. On the other hand, one cannot use the usual UV normalizability condition since there is a Landau pole which severely modifies the UV behavior. As anticipated in section 2, the natural condition is to require for each excitation that:

(3.21)

These conditions amount to introducing by hand an infinite wall at . Our working prescription will be to set , i.e. we consider a UV completion of the theory slightly below the scale in which the holographic -function becomes singular.

We now turn to justifying the proviso (3.21). Schrödinger potentials associated with the fluctuation equations can be computed following the definitions of appendix B. Figure 3 shows two examples of Schrödinger potentials for the vector excitation. For the scalar excitation, the plots are qualitatively similar (except that diverges towards in the UV pathological region).

Figure 3: On the left, the Schrödinger potential for the vector excitation with , . The UV pathological region is well separated from the IR region. On the right, the potential for , . Obviously, the UV region is not separated from the IR one. The potentials diverge to minus infinity at .

From the figure, it is apparent that as long as is large ( is small), the UV pathological region lies very far from the minimum of the potential where the physical wave function has its main support (see the plot on the left). On general grounds, we expect that whatever would be the UV completion of the potential, it only affects the eigenvalues by exponentially suppressed quantities. In particular, the precise value of has only a negligible effect on the spectrum, as long as lies far away from the IR region where the potential reaches its minimum.

On the other hand, if is too large, (and thus ) is not far from the IR region. Then, the formalism breaks down. In physical terms, if the UV completion sets in not far from the relevant IR scale, it will affect the IR physics in a non-negligible way. The plot on the right of figure 3 shows an example of this behavior. For this reason, we will restrict ourselves to .

3.3.2 Estimates of the meson spectrum from WKB

We have checked that the WKB expressions (3.17) in the unquenched case are also in very good agreement with the numerical values obtained from the shooting technique.777The relative error of the (3.17) formula with respect to the obtained numerical values is always well below , except for , when in the quenched case the error is already and in the unquenched cases remains approximately of the same order. The same applies to the case of unquenched massive flavors to be studied in the next section. Thus, we just concentrate on studying the WKB integrals (B.16), (B.17).

For small (large ), we can expand the integrands in (B.16), (B.17). A simple computation shows that:

(3.22)

It is clear from (3.22) that the leading correction to the masses, due to the dynamical flavors, is small. It is also approximately linear in , see the bottom lines in figure 4. The fact that the meson masses decrease with might appear counter-intuitive at first sight. In fact, there is nothing strange about it, as we will argue in the subsection 3.3.3. Moreover, a qualitatively similar behavior can be inferred from the unquenched lattice calculation reported in [19].

Now that we have found (3.22), let us be more precise about the size of the uncertainties of the spectrum associated to UV prescriptions. For large (but ), the integrand of the WKB integrals (B.16), (B.17) behaves as . Thus, if one takes a different value of , the variation of the integral and thus of the masses is of order . Taking into account the radius energy relation (where is the energy scale), one sees that different UV completions yield variations of the meson masses of order , where the IR scale is here associated to the mass of the quark constituents of the meson and the scale at which the UV completion sets in. Notice that these corrections are far more suppressed than the first corrections displayed in (3.22), which are of order (logarithmic in the energy scale).

3.3.3 Comments on the physical interpretation of the results

From eqs. (3.8), (3.17), (3.22), we read the following expression for the tower of vector mesons:

(3.23)

From this formula, it seems that meson masses decrease with , what, in turn, seems to contradict the fact that dynamical quarks screen the color charges: the binding energy between quark and anti-quark should decrease with increasing and, accordingly, meson masses should increase. The crucial point is that the flavor effects on the meson masses heavily depend on the scales we keep fixed while comparing different theories. This means that (3.23) states that meson masses decrease with if the IR quantities and are kept fixed when comparing theories with different . Had we kept fixed the gauge coupling at some UV scale, larger would have yielded smaller , possibly resulting in larger meson masses. We now clarify this statement with an example.

Suppose we want to rewrite (3.23) in terms of and of the ’t Hooft coupling at a scale . Due to the Landau pole, it does not make sense to take in the far UV, and in fact let us assume . Using (3.9), (3.19), (3.20), we find that:

(3.24)

Equation (3.23) can be rewritten as:

(3.25)

Thus, when we compare theories with different keeping fixed and the coupling at any scale , masses do indeed increase with .

Analogous considerations apply to the scalar mesons and to the rest of the cases discussed later in the paper as well.

3.4 Mesons in the presence of unquenched massive flavors

We can generalize the analysis of the previous section to the case in which the backreacting dynamical flavors are massive. We take these dynamical flavors to be of the same (non-chiral) type as the probe one, but with different mass (such that the tip of the backreacting branes lies at ). The background geometry is obtained by inserting in (3.4) the expression for computed in [7], namely for and for . Let us define:

(3.26)

so the functions determining the background can be written as:

(3.27)

and:

(3.28)

With this input, we can easily analyze the WKB integrals (B.16), (B.17). Now, they depend on two parameters, namely and the mass of the dynamical quarks through the quantity . Results are plotted in figure 4. At fixed , the meson masses slightly decrease with . As , the ’s tend to the quenched values (3.18). Moreover, the larger the value of , the larger are the meson masses at fixed . For large , the ’s tend to the quenched values too, i.e. the lines in figure 4 become horizontal. This is expected since if the dynamical quarks are very massive, they become quenched, even if there are many of them. At small , one recovers the massless quark set-up of section 3.3 and indeed the lower line in each plot is well approximated by (3.22).

Figure 4: The functions entering the meson mass formulae, see eqs. (3.17), (3.8). In each plot, the different lines are for a series of values of , namely .

3.5 On the constituent quark mass

As explained above, the definition (3.8), which uses the local string tension, and (3.15), which involves the constituent mass defined as in (3.14), coincide in the quenched (conformal) case, but differ in the non-conformal cases. We study here how the dimensionful prefactors in these equations vary with respect to each other when varying . Let us define the following quantity:

(3.29)

By Taylor expanding for large (small ), we find the following leading order expression for :

(3.30)

where we have defined . Both for or , one has , recovering the quenched result. This relation, together with the results in figure 4, allows one to compute how the ’s would vary with if, instead of (3.8) one decided to use (3.14), (3.15) as the definition of .

3.6 The holographic -function in flavored KW models

As we have anticipated in section 2, the D3-D7 conifold models under study are plagued by various UV pathologies. The behavior of the holographic -function (see eq. (2.4)), in particular, suggests that any UV cutoff we choose to adopt in the analysis of the mesonic spectra has to be located below the discontinuity point where the function diverges. Fixing the overall constant factor such that in the unflavored case, the function for the flavored KW models can be taken as:

(3.31)

where

(3.32)

Let us now consider the flavored KW model with massless quarks. The function , which in the unflavored case is monotonically increasing as , now has a maximum at .888 starts from zero at and comes backs to zero at . This behavior does not depend on the particular choice of integration constant for the warp factor. This behavior strongly affects that of the holographic -function999It is simple to realize that does not depend on and on the integration constant . It just depends on and is proportional to . as it is evident in figure 5.

Figure 5: The holographic -function in the massless-flavored KW model as a function of (left) and of (right).

As expected, the holographic -function has a bad discontinuity at , where . For , is positive and increasing with , whereas for it grows from minus infinity to zero.

Far below the Landau pole, the holographic -function is only slightly varying with and has an almost linear behavior

(3.33)

where . This expression can be read as the leading correction to the effective number of adjoint plus bifundamental degrees of freedom, due to the internal quark loops (see analogous comments for the entropy of the D3-D7 model in flat space in ref. [20]).

For the non-chiral massive-flavored KW solution, where the sea quarks have a mass which is related to , relevant plots are shown in figure 6.

Figure 6: The holographic -function in the non chiral massive flavored KW model. We have fixed . On the right, a zoom around .

The holographic -function approaches the unflavored value , for ; then it is very slowly increasing with up to . Slightly above the function starts increasing faster and then it blows up towards a point where the previously noted bad discontinuity shows up. In the present case is a function of : it goes to , as in the massless flavored case, for very small sea quark masses () and goes to for very large masses. For the -function is again going from minus infinity to zero.

In section 5.2, we will suggest a possible relation between the behavior of the -function and that of the functions in the mesonic spectrum.

4 Meson excitations in the flavored KS models

This section follows the same structure as the previous one, but we consider solutions in which the conifold metric is deformed and there are three-form fluxes that affect the UV behavior of the background. Namely, the quenched set-up is the Klebanov-Strassler solution [2]. We discuss mesonic excitations in this background and in its generalizations with unquenched massless [4] and massive [8] non-chiral flavors. The solution with massless chiral flavors was found in [5], but we will not address that case here.

4.1 Background and excitation equations

Let us write the metric in the notation of [16]. In Einstein frame it reads (both for the quenched and unquenched cases):

(4.1)

where we have used the definitions in (A.9). There are non-trivial , , , forms, which we will not need in the following. We refer the reader to [4] for details.

For the fluctuating probe brane, we will consider a supersymmetric embedding first discussed in [16], where it was shown to correspond to a massless non-chiral flavor in this set-up. Concretely, the embedding is parameterized as , , in terms of the variables introduced in appendix A. We discuss in the following mesonic excitations of this probe brane. Note that this simple massless embedding cannot be used in the KW cases because it would not produce a discrete spectrum, due to the different IR behavior of the backgrounds. Oscillations around massive generalizations of this embedding (in the quenched background) have been discussed in [21], but are beyond the scope of the present work.

Similarly to (3.5), let us just consider excitations of the gauge field of the form:

(4.2)

The corresponding second order equations are [16]:101010In [16], a constant dilaton was considered but it turns out that a running dilaton does not modify the expressions (4.3). In the very far UV these equations coincide with those of KW, as expected, see eqs. (3.6), (3.7). Looking at the far UV amounts to setting , . One also has to identify , and .