A Derivation on the quadratic fluctuation equations

The discontinuities of conformal transitions and mass spectra of V-QCD

Abstract:

Zero temperature spectra of mesons and glueballs are analyzed in a class of holographic bottom-up models for QCD in the Veneziano limit, , , with fixed (V-QCD). The backreaction of flavor on color is fully included. It is found that spectra are discrete and gapped (modulo the pions) in the QCD regime, for below the critical value where the conformal transition takes place. The masses uniformly converge to zero in the walking region due to Miransky scaling. All the ratios of masses asymptote to non-zero constants as and therefore there is no “dilaton” in the spectrum. The S-parameter is computed and found to be of in units of in the walking regime, while it is always an increasing function of . This indicates the presence of a subtle discontinuity of correlation functions across the conformal transition at .

1

1 Introduction and Outlook

The gauge/gravity duality has been widely used to describe the physics of strongly coupled gauge theories. The perturbative expansion of Yang-Mills (YM) theory in the (’t Hooft) large limit, and , suggests that gauge theories have a dual string theory description. We will consider QCD with gauge group and with quarks in the fundamental representation of . In the limit of large , the effects of flavor degrees of freedom are suppressed. Veneziano introduced an alternative topological expansion of QCD, via the so-called Veneziano limit

(1.1)

in this limit we may study phenomena depending on the flavor degrees of freedom such as hadronic multiparticle production, [1] and the problem in QCD, [2].

By studying the QCD -function,

(1.2)

with

(1.3)

it is evident that the theory in the Veneziano limit exhibits several interesting features.

  • For the theory is not asymptotically free but IR free, and therefore the IR physics is simple to assess.

  • At values , there is also an IR fixed point of the two-loop -function. This region of where the theory has an IR fixed point is called “conformal window”. For close to (Banks-Zaks region) the IR fixed point is at weak coupling, [3]. In the Banks-Zaks region, perturbation theory is trustworthy at all energy scales. As decreases, the IR fixed point coupling increases and perturbative methods cannot be applied. QCD with in the conformal window has unbroken chiral symmetry.

  • The theory is expected to reduce (at qualitative level) to standard QCD for small and to pure Yang-Mills for . Therefore, there should be a critical value where there is a transition from chirally symmetric theories in the conformal window to confining theories with broken chiral symmetry in the IR. In the region below but close to the theory is expected to exhibit “walking” behavior. There is an approximate IR fixed point in the RG flow, which dominates the behavior of the theory for a large range of energies. It is eventually missed by the RG flow and the theory ends up in the IR regime of broken chiral symmetry. In particular, the coupling constant varies slowly for a long energy range, [4].

The standard picture is that the transition at is a phase transition of the Berezinskii-Kosterlitz-Thouless (BKT) type, [5, 6] and is known as a conformal phase transition, [6]. It has been proposed to be caused by the annihilation of an IR and a UV fixed point, [7]. The dimensionful observables of the theory scale as the condensate in the BKT transition in two dimensions, (also known in this context as Miransky scaling). The value of is difficult to be determined since the dynamics of the theory is strongly coupled in this region. It is also difficult to disentangle this transition in lattice calculation due to finite volume limitations. There have, however, been many efforts along various directions in order to find the critical for finite where the transition takes place, see [8], [9], [10] and [11].

Nearly conformal gauge theories are of great importance in models for the physics beyond the Standard Model, [12], [13]. In particular, they are an important ingredient of technicolor models where electroweak symmetry breaks spontaneously through the same mechanism as chiral symmetry in QCD. However, in technicolor the breaking happens at a higher energy scale and in a new gauge sector, [14], [15]. The model includes “technifermions” transforming in some representation of the technicolor group whose condensate breaks dynamically the electroweak symmetry.

In order to generate the correct masses of the Standard model fermions, technicolor is generalized to extended technicolor which has a larger hidden gauge group. It follows that the lepton and quark masses depend on the dimension of the scalar operator that condenses. For phenomenological purposes (related to the size of the lepton and quark masses) the dimension of this operator should be away from three, which is its free field theory value, and should be closer to two. This can be achieved in a “walking” theory, [8], [16] and [17]. A typical issue with technicolor models is that they add relatively large contributions to the S-parameter, which was first defined in [18]. Such contributions create tension with the electroweak precision measurements. Walking theories have been conjectured to have reduced, or even arbitrarily small S-parameter, [19].

Several efforts have been made to study gauge theories including flavor degrees of freedom in the context of AdS/CFT. Most of the research has been concentrated in the case of the quenched approximation, , where flavor branes are introduced in the color background geometry without backreaction, [20]. In order to study the Veneziano limit of gauge theories holographically, the backreaction of flavor to the color action should be taken into account. A possible way of attacking such a problem is to consider the action of type IIA/B supergravity with smeared flavor branes, [21, 22]. Alternatively, the flavor sources may be localized, so their charge density is a sum of delta functions. From the string theory viewpoint, such systems correspond to intersecting branes. Some well known examples are the D3-D7 intersections, [23], D2-D6, [24], D4-D8, [25], and D4-D8-, [26].

Bottom-up holographic models, which are based on the hard-wall AdS/QCD model [27], have been constructed to describe walking field theories, [28, 29]. Top-down approaches for describing field theories with walking behavior are [30] and [31].

In [4], a bottom-up class of holographic models was built for QCD in the Veneziano limit, called V-QCD. The sector of QCD was described by the Improved Holographic QCD (IHQCD) model, [32], whose action is the Einstein-dilaton one with a specific, non-trivial dilaton potential. The dilaton is the field dual to the scalar operator of YM. The near-boundary asymptotics of the potential are chosen in order to match the perturbative -function of Yang-Mills coupling and the IR asymptotics are such that reproduce features as confinement, asymptotic linear glueball trajectories and a mass gap. By tuning two phenomenological parameters of the potential, the model agrees with lattice data both at zero and finite temperature, [33], [34]. For a review of lattice studies of large gauge theories see [35]. There have been efforts to model walking behavior in the context of a single scalar model by adjusting appropriately the -function (potential), [10], [36]. However, in those cases no flavor degrees of freedom were considered.

In [4] fully backreacting flavor degrees of freedom were included in the IHQCD backgrounds. The framework for this class of models model was first studied in [21, 37]. Its flavor action is the low-energy effective action of brane-antibrane pairs. This action was first proposed by Sen, [38] around flat spacetime. The fields of the model include the complex tachyon which corresponds to the lightest state of the string stretching among branes and antibranes, as well as a left and a right gauge field dual to the left and right flavor QCD currents, respectively. The tachyon field is dual to the quark mass operator whose non-trivial vacuum expectation value causes chiral symmetry breaking . The model was found to reproduce various low energy features of the QCD meson sector, [39]. Tachyon condensation has also been studied in the Sakai-Sugimoto model, [40]. A lattice study of meson physics of large gauge theories was published recently in [41].

V-QCD is therefore created by the fusion of IHQCD and tachyon dynamics, as modeled by generalizations of the Sen action. The dilaton potential is taken to have the same form as in the IHQCD setup. The tachyon potential must have two basic properties; to vanish exponentially in the IR in order to have the brane-antibrane pair annihilate and to give the right UV mass dimension to the dual operator of the tachyon.

The vacuum saddle point of the theory is determined by the non-trivial profiles of the metric, dilaton and tachyon fields. The left and right gauge fields are trivial in the vacuum solution. Making a few reasonable assumptions, the model produces a phase diagram which has similar structure as that expected from QCD in the Veneziano limit and does not depend on the details of the potentials.

In the range , where the theory has an IR fixed point, the IR dimension of the chiral condensate can be determined. It is found to decrease as a function of and the point where it becomes 2, determines , as has been argued in [7]. Upon matching the -function of the Yang-Mills coupling and the anomalous dimension of to the QCD result in the UV, is found to be close to 4,

(1.4)

which agrees with other estimates, [8, 9, 10]. A similar phase transition has also been found in more simplified holographic models for QCD, which were likewise matched properties of perturbative QCD near the boundary, [42].

V-QCD models have been analyzed in detail at finite temperature in [43]. Generically they exhibited a non-trivial chiral-restoration transition in the QCD phase and no transition in the conformal window. Depending on the model class, they might exhibit more than one phase transitions especially in the walking region.

In the present article, we study the quadratic fluctuations of V-QCD for zero quark masses. Some of the main results first appeared in the short publication [44]. Among other issues, there are two relevant questions for walking gauge theories which we will answer here:

  • Is there a light dilaton2 state in the spectrum, due to the approximate scale invariance, [13]?

  • Is the S-parameter strongly suppressed as expected on general grounds, [19]?

Various holographic models have been proposed which explore the above questions. In [45] and [31] the lightest state is found to be a scalar. But its identification with the dilaton seems to be model dependent. The S-parameter has also been calculated in holographic bottom-up, [29], and Sakai-Sugimoto like models, [46]. In [47], it was argued that the S-parameter is bounded below for a specific class of holographic models.

We consider small fluctuations of the fields which are involved in the action. The glue sector has the metric, the dilaton and the (closed string) axion. Their normalizable fluctuations correspond to glueballs, where is the spin, is parity and the charge conjugation of the state. The flavor action includes the complex tachyon and the left and right gauge fields. These give rise to towers of mesons. The states above are separated in two distinct symmetry classes, the flavor non-singlet and the flavor singlet states. Singlet or non-singlet fluctuations with different have an infinite tower of excited states. The flavor singlet states which are in the meson and in the glueball sectors mix at leading order in , since we explore the Veneziano limit (such a mixing is in the ’t Hooft limit).

As mentioned above, the dilaton and the tachyon potentials are constrained by QCD properties, like confinement, glueball spectra, anomalies etc. Meson spectra which are calculated in this work also set some constraints on the potentials which are used in the action, see section 4.4. We explore the correlation between different IR asymptotics of the various potential functions that enter the holographic V-QCD action, and properties of the glueball and meson spectra. Once this is done we can pin down the asymptotics that provide the correct expected gross properties. The main requirements in the flavor non-singlet sector are that all meson towers have linear asymptotic (radial) trajectories. In the flavor singlet case, mesons and glueballs decouple at large excitation number and the mesons have the same asymptotics as the flavor non-singlet meson with the same . The glueball trajectories are linear. Even if the above requirements do not set tight constraints on the potentials appearing in the action, they constrain the region of their parameters, see section 4.4.

An interesting related issue is whether the slopes of the linear trajectories for the axial vector and the vector mesons are the same. It has been pointed out in [37] that, due to chiral symmetry breaking, they might differ. Indeed in the models discussed in [37] and [39] these slopes were different. Despite an ensuing debate in the literature, (see [48]-[50] and references therein), what happens in QCD remains an open issue. We will also characterize this possibility in terms of the IR asymptotics of the flavor potential functions.

The numerical solution of the fluctuation equations for zero quark mass produces the spectrum of the model, as described in section 5. The analysis of the spectrum was done for two different classes of tachyon potentials, potentials I and potentials II. Potentials I reproduce well the physics of real QCD in the Veneziano limit. For instance, it has been checked that the finite temperature phase diagram has the correct structure, [43]. The asymptotic meson trajectories are also linear but with possible logarithmic corrections. Potentials II are a bit further from the detailed QCD behavior. We have investigated them in order to study the robustness of our results against changes in the asymptotic form of the potential. Potentials II have quadratic asymptotic trajectories for mesons.

1.1 Results

We summarize below our results.

  1. The main generic properties of the spectra are as follows.

    • Below the conformal window, in the chirally broken phase with , the spectra are discrete and gapped. The only exception are the pseudoscalar pions that are massless, due to chiral symmetry breaking.

    • In the conformal window, , all spectra are continuous and gapless.

    • All masses in the Miransky scaling region (aka “walking region”) are obeying Miransky scaling . This is explicitly seen in the case of the mass in figure 5.

  2. The non-singlet fluctuations include the L and R vector meson fluctuations, packaged into an axial and vector basis, , the pseudoscalar mesons (including the massless pions), and the scalar mesons. Their second order equations are relatively simple. Our main results for the non-singlet sector are:

    • The mass spectra of the low lying mesons can be seen in figure 3 for potentials I and in figure 4 for potentials II (note that the left hand plots have their vertical axis in logarithmic scale). The lowest masses of the mesons vary little with until we reach the walking region. There, Miransky scaling takes over and the masses dip down exponentially fast. The scale is extracted as usual from the logarithmic running of in the UV.

    • The mass ratios asymptote to finite and constants as .

  3. The singlet fluctuations include the glueballs, the glueballs and scalar mesons that mix to leading order in in the Veneziano limit, and the glueballs and the pseudoscalar tower. Although the spin-two fluctuation equations are always simple, summarized by the appropriate Laplacian, the scalar and pseudoscalar equations are very involved. Our main results for the singlet sector are:

    • The anomaly appears at leading order and the mixture of the glueball and the has a mass of .

    • In figures 6 and 7 we present the results for the singlet scalar meson spectra. The dependence on is qualitatively similar as in the non-singlet sector.

    • In the scalar sector, for small , where the mixing between glueballs and mesons is small, the lightest state is a meson, the next lightest state is a glueball, the next a meson and so on. However, with increasing , non-trivial mixing sets in and level-crossing seems to be generic. This can be seen in figure 7.

    • All singlet mass ratios asymptote to constants as (see figure 6). The same holds for mass ratios between the flavor singlet and non-singlet sectors, as confirmed numerically in figure 8. There seems to be no unusually light state (termed the “dilaton”) that reflects the nearly unbroken scale invariance in the walking region. The reason is a posteriori simple: the nearly unbroken scale invariance is reflected in the whole spectrum of bound states scaling exponentially to zero due to Miransky scaling. The breaking however of the scale invariance is not spontaneous.

  4. The asymptotics of the spectra at high masses is in general a power-law with logarithmic corrections, with the powers depending on the potentials. The trajectories are approximately linear () for type I potentials and quadratic () for type II potentials. There is the possibility, first seen in [37] that the proportionality coefficient in the linear case is different between axial and vector mesons.

  5. There are several dilaton-dependent functions that enter the V-QCD action, which can be constrained by using various known properties of QCD. They include the dilaton potential in the glue sector action in (2.18), as well as the tachyon potential , the kinetic function for the tachyon, , and the kinetic function for the gauge fields that appear in the flavor action (2.20). Moreover, the tachyon potential function, motivated by flat space string theory, is parametrized as in (2.25)

    (1.5)

    in terms of two extra functions of the dilaton, and . We find that the functions can be constrained as follows.

    • The UV asymptotics of these functions can be fixed from perturbation theory. The IR asymptotics are more obscure. The pure glue potential has been established in previous works, [32, 34].

    • We have parametrized the IR asymptotics as

      (1.6)

      is constrained indirectly so that has a non-trivial IR fixed point for a range of . The others however are severely constrained. By asking various generic criteria to be satisfied as well as requiring the existence of asymptotically linear meson towers, we obtain that

      (1.7)

      and

      (1.8)

      If on the other hand constant, then the axial and vector towers have different slopes. In view of this we opt for

      (1.9)

      These parameter values are essentially those of potentials I.

      It is curious to note that all the functions, at large behave (modulo the logs) like in standard non-critical tree-level string theory, (in Einstein frame).

  6. In the region and for zero quark mass, there are infinite subdominant saddle points, that we called Efimov solutions and which are labeled by a natural enumeration , that indicates the number of zeros of the tachyon solution, [4]. We have verified numerically and analytically3 that these saddle points are perturbatively unstable. Tachyonic fluctuation modes are seen in the scalar singlet and non-singlet towers.

  7. The behavior of correlation functions across the conformal transition turned out to be interesting and in part different from previous expectations. We have computed the two-point functions of several operators including the axial and vector currents. We focus on the two-point function of the vector and axial currents which can be written in momentum space as

    (1.10)

    and similarly for the axial vector. We have the decomposition

    (1.11)

    where , are the flavor group generators and are the radial wave-functions.

    • Using the expansions

      (1.12)

      we determine as

      (1.13)

      where the normalization was fixed by matching the UV limit of the two-point functions to QCD. The dependence of on is shown in figure 9. The pion decay constant changes smoothly for most , but is affected directly by Miransky scaling which makes it vanish exponentially in the walking regime.

    • The S-parameter is defined as

      (1.14)

      As both masses and decay constants in (1.12,1.14) are affected similarly by Miransky scaling, the S-parameter is insensitive to it (Miransky-scaling-invariant). Therefore its value cannot be predicted by Miransky scaling alone. Our results show that generically the S-parameter (in units of ) remains finite in the QCD regime, and asymptotes to a finite constant at (see figure 10). The S-parameter is identically zero inside the conformal window (massless quarks) because of unbroken chiral symmetry. This suggests a subtle discontinuity of correlators across the conformal transition. We have also found choices of potentials where the S-parameter becomes very large as we approach . Our most important result is that generically the S-parameter is an increasing function of , reaching it highest value at contrary to previous expectations, [51].

    • We have also calculated the next derivative of the difference, (related to the X-parameter of [52]) as

      (1.15)

      so that

      (1.16)

      This parameter is shown for both potentials I and II in Fig. 11. The dependence (in IR units) is qualitatively rather similar to the S-parameter so that the values typically increase with , and approach fixed values as . However, unlike for the S-parameter, there is also a region with decreasing values near .

1.2 Outlook

Our exhaustive analysis of the class of V-QCD models and the results obtained paint a reasonably clear holographic picture for the behavior of QCD in the Veneziano limit. Although V-QCD should be considered as a toy model for QCD in the Veneziano limit, there are two facts that give substantial weight to our findings.

  • The ingredients of the holographic models follow as closely as possible what we know from string theory about the dynamics of the dilaton and open-string tachyons. This is treated in more detail in section 2.4.

  • We have explored parametrizations of the functions and potentials that enter the holographic action, especially in the IR. General qualitative guidelines suggest that these functions are the same as in (naive) string theory corrected by logarithms of the string coupling. This was first seen in [32] where the dilaton potential behaves as as . Note that in the Einstein frame the noncritical dilaton potential in five dimensions is proportional to .

    Moreover the power of the subleading log was fixed at the time in order for the glueball radial trajectories to be linear. It was later realized that only asymptotic potential of the form lead to non scale-invariant IR asymptotics, [53]. It was also independently found, [54], that the power was also responsible for providing the well known power of in the free energy just above the deconfinement phase transition, [55].

Despite the success of the framework, there are several conceptual issues that remain to be addressed successfully.

  • The effects of loops of the non-singlet mesons are not suppressed, as the large number of flavors compensates for the large suppression of interactions.

  • The CP-odd sector requires further attention as the limit in that sector seems to not be smooth.

To this we add two obvious open problems that involve the understanding of phase diagram at finite temperature and density, and the construction of a well-tuned model to real QCD. All of the above are under current scrutiny.

2 V-Qcd

The complete action for the V-QCD model can be written as

(2.17)

where , , and are the actions for the glue, flavor and CP-odd sectors, respectively. We will define these three terms separately below. As discussed in [4], only the first two terms contribute in the vacuum structure of the theory if the phases of the quark mass matrix and the angle vanish. The full structure of and was not detailed in [4], as this was not necessary in order to study the vacuum structure of the model. However, the extra terms do contribute to the spectrum of fluctuations and will therefore be discussed in detail below.

2.1 The glue sector

The glue action was introduced in [32],

(2.18)

Here is the exponential of the dilaton. It is dual to the operator, and its background value is identified as the ’t Hooft coupling. The Ansatz for the vacuum solution of the metric is

(2.19)

where the warp factor is identified as the logarithm of the energy scale in field theory. Our convention will be that the UV boundary lies at (and ), and the bulk coordinate therefore runs from zero to infinity. The metric will be close to the AdS one except near the IR singularity at . Consequently, , where is the (UV) AdS radius. In the UV is therefore identified roughly as the inverse of the energy scale of the dual field theory.

2.2 The flavor sector

The flavor action is the generalized Sen’s action,

(2.20)

where the quantities inside the square roots are defined as

(2.21)

with the covariant derivative

(2.22)

The fields , as well as are matrices in the flavor space. We also define

(2.23)

It is not known, in general, how the determinants over the Lorentz indices in (2.20) should be defined when the arguments (2.21) contain non-Abelian matrices in flavor space. However, for our purposes such definition is not required: our background solution will be proportional to the unit matrix , as the quarks will be all massless or all have the same mass . In such a case, the fluctuations of the Lagrangian are unambiguous up to quadratic order.

The form of the tachyon potential that we will use for the derivation of the spectra is

(2.24)

This is the string theory tachyon potential where the constants have been allowed to depend on the dilaton . For the vacuum solutions (with flavor independent quark mass) we will have where is real, so that

(2.25)

The coupling functions and are allowed in general to depend on , through such combinations that the expressions (2.21) transform covariantly under flavor symmetry. In this paper, we will take them to be eventually independent of , emulating the known string theory results.

We discuss how the -dependent functions , , , and should be chosen in Section 4.

2.3 The CP-odd sector and the closed string axion

Here we follow [37] in order to discuss the coupling of the closed string axion to the phase of the bifundamental tachyon, dual to the quark mass operator and the axial U(1) gauge boson. This discussion adapted to 5d holographic QCD is as follows.

We start with a three-form RR axion , with field strength, and

(2.26)

and

(2.27)

Here is the overall phase of the tachyon, . In flat-space tachyon condensation is independent of the dilaton, and is the same as the potential that appears in the tachyon DBI, [37]. In our case it may be different in principle. However, it must have the same basic properties; in particular it becomes a field-independent constant (related to the anomaly) at , and vanishes exponentially at . We may dualize the three-form to a pseudo-scalar axion field by solving the equations of motion as

(2.28)

Therefore, the dual action takes the form

(2.29)

in terms of the QCD axion .

This is normalized so that is dual to with being the standard -angle of QCD. The coupling to the axial vector, , reflects the axial anomaly in QCD

(2.30)

with , which gives the correct U(1) anomaly. This normalization is correct when is normalized so that the two-point function of the dual current

From the coupling between source and operator, , with we obtain the parametrization .

The terms above mix the axion both with and the longitudinal part of . As we will see, for and there are other terms coming from the DBI action. In the ’t Hooft limit, and the flavor corrections are subleading, whereas in the Veneziano limit , and the corrections are important.

The natural form for the function is to keep the tachyon exponential without the term, i.e., . We will work out the fluctuation problem however for general .

Since we have in the background, we must first solve the action to determine and . In the quenched limit, the calculation we have done is enough to leading order in . In the Veneziano limit the full second order fluctuation system must be derived.

2.4 The relation to string theory models

We would like here to compare the class of models we are studying in this paper to expectations from string theory and QCD.

The main ingredients associated with the pure glue part of the model has been studied extensively and motivated from string theory. These discussions can be found in the original papers, [32]. It should be noted that the important region for such models is in the IR. Although a gravity description is not expected in the UV (due to the weak coupling), the approach taken in [32] and here is that of matching the gravitational theory to reproduce perturbative -functions in the UV. This guarantees the correct UV boundary conditions for the (more interesting) IR theory.

In the flavor sector, the main ingredient is the Sen-like action for open string tachyons of unstable branes and brane-antibrane pairs in flat space, [38]. Although this action has not been tested in all possible contexts it has passed a lot of non-trivial tests in the past.

However, it is not immediately obvious how the action should be written down in the case of a curved background and a running dilaton that is relevant here. There are nevertheless some simplifications:

  • All YM-like geometries in the IR that have been studied in the past, [32], are nearly flat in the string frame. This is non-trivial and we have no deep understanding of this fact. There are hints though: the asymptotic potentials in the Einstein frame

    (2.31)

    become in the string frame. The case corresponds to the non-critical string dilaton potential that is constant in the string frame. It gives rise to the well-known linear dilaton solution where the dilaton runs but the string metric remains flat. The solutions for positive are similar. The metric is almost flat and the dilaton runs as the power of the radial coordinate. is the case relevant for QCD, as it is the only one that gives linear asymptotic trajectories for glueballs, and the appropriate temperature scaling of the free energy just above the phase transition.

  • Given this, it is a natural choice to use an adiabatic Ansatz for the flavor action: to make all constants in the Sen action dilaton dependent. This is precisely what we do.

  • Very little is known about the non-abelian tachyon action. This however is not limiting our analysis provided (a) we do not break the vector symmetry, and (b) we are studying effects up to second order in the fluctuations. If the vector symmetry is not broken by the quark masses, the vacuum expectation value for the tachyon is proportional to the unit matrix, and therefore non-abelian subtleties are absent. Similarly, up to quadratic order in fluctuations, the non-abelian subtleties do not arise if we use for example a symmetric prescription.

  • The Sen’s action, like any other DBI-like action, has its limitations; it does not include “acceleration corrections” (terms ). These are well defined in the abelian (flavor) case, but apply also to the non-abelian case relevant for our purposes. It is therefore important to check that these corrections are not important for our solutions.

    Indeed an analysis directly in the string frame of all tachyon solutions for all potentials used here, shows they have “small accelerations” in the IR regime. This might not be so in the middle of the geometry. However that regime is not important for the physics. The only exception to the above is the “walking regime”, for smaller and close to ; yet we have checked that for that case too, the accelerations remain small.

  • Unlike the ’t Hooft limit, where the loops of singlet bulk fields are suppressed by extra powers of , in the Veneziano limit, there is no such suppression for the non-singlet flavor fields. Therefore, we would have to think of our bottom-up action as the Wilsonian effective action that includes the effects of integrating out massive modes. In the QCD phase, it therefore does not contain the contributions of the pions which are the only massless states. Although some contributions may be relevant in some cases, we do not expect them to modify substantially neither the vacuum structure, nor the finite temperature and density structure. This however needs further analysis.

    In the conformal window all modes are gapless. However, we do not expect instabilities of the Wilsonian action beyond the one specified by the only relevant operator (quark mass).

    The only subtle region is the one near the conformal transition. In that region qualitative changes may happen because of the quantum effects. This issue requires further analysis but we will not pursue it in this paper.

  • Within the framework set by the general principles above, we have analyzed many different types of IR actions (especially their dependence on the dilaton). We have mapped their IR landscape onto important phenomenological properties of the theory. This has been analyzed in section 4.2 and appendices D, E and G.

    The remarkable conclusion is that, like the glue part of the theory, the dilaton functions, apart from logarithmic corrections, should have in the IR the same values as in standard non-critical string theory around flat space. For the dilaton functions and , defined in (2.21) and (2.25) we obtain from (5) (and dropping the logarithms)

    (2.32)

    and after transforming to the string frame they become

    (2.33)

    in agreement with tree level string theory results.

    It should be also mentioned that the parameter least sensitive to phenomenological constraints is the power of . Indeed our potentials I, which were constructed such that all constraints from QCD are satisfied at qualitative level, implement the choice of (2.4) except for the value of (see subsection 4.5). We chose a polynomial and therefore for simplicity, but modifying the power to would not result in any qualitative changes.

  • Another issue is the justification of the use of the DBI form of the tachyon action even after integrating out non-singlet degrees of freedom. This issue turns out to not be important in the intermediate regions of the geometry, as the results depend very little on this region.

    The only regime where this issue is of importance is when the kinetic term of the tachyon, , diverges. This is happening only in the IR part of the geometry, if the tachyon condenses and if is constant (as is the case for potentials I).

    We have analyzed various asymptotic powers in the actions different from the square root characteristic of DBI by parametrizing the tachyon action as as with . This is discussed in Appendix D.2.3. We have found that is a “critical value” for the exponent where many cancelations are operative in the equations of motion. For there are no regular solutions to the equations of motion, whereas for the diverging tachyon solutions are powerlike rather than exponential. Meson trajectories in this case cannot be linear.

    These are indirect but convincing arguments for the use of tachyon DBI action, but a first principles derivation in string theory is however desirable.

To conclude this section, V-QCD is a toy model for real QCD but it seems to have many features that suggest it belongs to the correct universality class.

2.5 The background solutions

We will discuss some general features of the background solutions of the V-QCD models. We restrict first to the standard case, which has a phase diagram similar to what is usually expected to arise in QCD. In Sec. 4 we shall discuss which V-QCD models fall in this category.

To find the background, we consider -dependent Ansätze for , and . Assuming that the quark mass is flavor independent, we further take , set all other fields to zero, and look for solutions to the equations of motion. The models are expected to have two types of (zero temperature) vacuum solutions [4]:

  1. Backgrounds with nontrivial , and with zero tachyon . These solutions have zero quark mass and intact chiral symmetry.

  2. Backgrounds with nontrivial , and . These solution have broken chiral symmetry. As usual, the quark mass and the chiral condensate are identified as the coefficients of the normalizable and non-normalizable tachyon modes in the UV (see Appendix D).

In the first case, the equations of motion can be integrated analytically into a single first order equation, which can easily be solved numerically. In the second case, we need to solve a set of coupled differential equations numerically. At the UV boundary and at the IR singularity, analytic expansions can be found (see Appendix D).

We constrain the ratio to the range where the upper bound was normalized to the Banks-Zaks (BZ) value in QCD, where the leading coefficient of the -function turns positive. The standard phase diagram at zero quark mass has two phases separated by a phase transition at some within this range.

  • When , chiral symmetry is intact. The dominant vacuum solution is of the first type with the tachyon vanishing identically.

  • When , chiral symmetry is broken. The dominant vacuum therefore has nonzero tachyon even though the quark mass is zero.

Interestingly, the phase transition at (which is only present at zero quark mass) involves BKT [5] or Miransky [6] scaling. The order parameter for the transition, the chiral condensate vanishes exponentially,

(2.34)

as from below. Here the constant is positive. When , is identically zero as chiral symmetry is intact. The Miransky scaling is linked to the “walking” behavior of the coupling constant: the field takes an approximately constant value for a wide range of , and the length of this region obeys the same scaling as (the square root of) the condensate in (2.34). The walking behavior is connected to the IR fixed point which is found for : then as , and the geometry becomes asymptotically AdS also in the IR.

The Miransky scaling can also be discussed in terms of the energy scales of the theory. We may define the UV and IR scales, denoted by and (see Appendix D for details). When and is not small, the V-QCD models involve only one scale, reflecting the behavior of ordinary QCD. We therefore have . When , the two scales become distinct, and their ratio obeys Miransky scaling:

(2.35)

The scale continues to be the one where the coupling constant becomes small even as . The coupling walks for , and starts to diverge at . In terms of the two scales, the chiral condensate behaves as . The result (2.34) is therefore understood to hold when is measured in the units of .

We recall how the constant can be evaluated (see Sec. 10 of [4] for details). Let be the bulk mass of the tachyon and the AdS radius, both evaluated at the IR fixed point. We then have

(2.36)

The models may also have subdominant vacua. Including the solutions with finite quark mass, the generic structure is as follows.

  • When , only one vacuum exists, even at finite quark mass.

  • When and the quark mass is zero, there is an infinite tower of (unstable) Efimov vacua in addition to the standard, dominant solution.4

  • When and the quark mass is nonzero, there is an even number (possibly zero) of Efimov vacua. The number of vacua increases with decreasing quark mass for fixed .

The infinite tower of Efimov vacua, which appears at zero quark mass, admits a natural enumeration (where is the number of tachyon nodes of the background solution). A generic feature of these backgrounds is, that they “walk” more than the dominant, standard vacuum, so that the scales and become well separated for all when is large enough. It is possible to show that

(2.37)

for any . Here can also be computed analytically (see Appendix F in [4]). On the other hand, as we find that

(2.38)

for any value of . In particular, corresponds to the standard solution of (2.35). We also found a similar scaling result for the free energies of the solutions as in [4], therefore proving that the Efimov vacua are indeed subdominant, and verified this numerically for all . In this article we shall show in Sec. 5 that the Efimov vacua are perturbatively unstable (again analytically as , and numerically for all ).

3 Quadratic fluctuations

In order to compute the spectrum of mesons and glueballs we need to study the fluctuations of all the fields of V-QCD. In the glue sector, the relevant fields are the metric , the dilaton and the QCD axion . Their normalizable fluctuations correspond to glueballs with , where stands for the spin and and for the field properties under parity and charge conjugation respectively. In the meson sector, one has the tachyon , and the gauge fields ; their normalizable fluctuations corresponding to mesons with .

The fluctuations fall into two classes according to their transformation properties under the flavor group: flavor non-singlet modes (expanded in terms of the generators of ) and flavor singlet modes. The glue sector contains only flavor singlet modes, whereas each fluctuation in the meson sector can be divided into flavor singlet and non-singlet terms. Those (flavor singlet) modes which are present in both sectors will mix. Since we are in the Veneziano limit, the mixing takes place at leading order in : the glueball mixes with the flavor singlet -meson, and the pseudoscalar flavor singlet meson mixes with the glueball due to the axial anomaly (realized by the CP-odd sector). All classes, with various and transformation properties under the group, contain an infinite discrete tower of excited states once we are below the conformal window.

To compute the masses of the different glueballs and mesons we expand the action up to quadratic order in the fluctuations and separate the fluctuations into the different decoupled sectors described above. We postpone the technical details of this analysis to Appendix A and in the rest of this section present the basic structure of each sector.

We start by defining the vector and axial vector combinations of the gauge fields:

(3.39)

They will appear both in the singlet and non-singlet flavor sectors that we describe in the following. We also write the complex tachyon field as

(3.40)

where are the generators of , is the background solution, () is the scalar (pseudoscalar) flavor singlet fluctuation, and () are the scalar (pseudoscalar) flavor non-singlet fluctuations.

3.1 The flavor non-singlet sector

The class of flavor non-singlet fluctuations involves the part of the vector, axial vector, pseudoscalar and scalar mesons. The relative fields are split as

(3.41)

where the superscript denotes that we are only considering the traceless terms of the fluctuations and . The gauge is chosen and and are transverse, . In addition, and are the fluctuations of the tachyon modulus and phase, respectively, as defined in (3.40).

The fluctuation equations are derived in Appendix A.1. The result for the vector and the transverse axial vector wave functions reads

(3.42)

where we introduced a shorthand notation for

(3.43)

and the various potentials were defined in sections 2.1 and 2.2. Notice that the two equations differ by a mass term which comes from the coupling of the axial vectors to the tachyon. This term implements the effect of chiral symmetry breaking, sourcing the differences between the spectra of vector and axial vector mesons.

The fluctuation equations for the non-singlet pseudoscalars and scalars are given in Eqs. (A.1.3) and (A.138), respectively, in Appendix A.1. The pseudoscalar fluctuations also contain the pions which become massless as the quark mass tends to zero, and obey the Gell-Mann-Oakes-Renner relation for small but finite quark mass.

3.2 The flavor singlet sector

We first consider the pseudoscalar fluctuations which give rise to the glueballs and the mesons. The flavor singlet pseudoscalar degrees of freedom correspond to gauge invariant combinations of the longitudinal part of the flavor singlet axial vector fluctuation , the pseudoscalar phase of the tachyon and the axion field . These fields are decomposed as

(3.44)

The gauge invariant combinations of the above fields are

(3.45)

which correspond to the pseudoscalar glueball () and meson towers. These combinations satisfy the coupled equations (A.194, LABEL:cpsys2), reflecting the expected mixing of the glueballs with the mesons.

The scalar fluctuations should realize the glueball and the flavor singlet meson. The contributing fields come from the expansion of the tachyon modulus (), of the dilaton () and the metric. The only scalar metric fluctuation which remains after eliminating the nondynamical degrees of freedom is , which appears as the coefficient of the flat Minkowski metric (see (A.104) and (A.170) in Appendix A). The combinations

(3.46)

are invariant under bulk diffeomorphisms (see section A.3.3) and correspond to scalar glueballs and mesons which mix at finite , see (A.203, A.204). As was first pointed out in [32] they correspond to RG invariant operators in the dual theory.

For the flavor singlet spin-one states there are no glueballs and hence no mixing. The singlet vectors have the same spectrum as the non-singlet ones. The fluctuation equation for the singlet axial vector fluctuations differs from the one for the non-singlet axials in (3.42) by a positive mass term, coming from the CP-odd part of the action, see Eq.(A.184). Therefore, the singlet axial vector states have generically higher masses than the non-singlet ones.

Finally, the singlet sector includes a traceless rank-2 tensor fluctuation, which satisfies the scalar Laplacian equation, (A.211), in the 5-dimensional background (2.19). This fluctuation generates the tower of glueballs.

4 Constraining the action

Agreement with the dynamics of QCD sets various requirements on the potential functions (, , , ) of V-QCD. In particular, both the UV and IR asymptotics of these functions need to fulfill constraints, which have been analyzed in part in [32, 4, 43]. In this article, we perform a more detailed analysis of the IR structure than was done before. In addition, we study the constraints arising from the meson spectra. These constraints also apply to the function , which has not been discussed in earlier work. We will first discuss generic features of the potentials, list the detailed constraints and give some examples below.

The asymptotics of the dilaton potential , which governs the glue dynamics, has been analyzed in detail in [32]. An overall fit to Yang-Mills data was done in [34].

There are other undetermined functions in the flavor action which can also depend on the tachyon field . When the quark mass is flavor independent, the background solution is of the form . Evaluated on the background, the potentials (, , ) must satisfy the following generic requirements [4]:

  • There should be two extrema in the potential for : an unstable maximum at with chiral symmetry intact and a minimum at with chiral symmetry broken.

  • The full dilaton potential at , namely , must have a nontrivial IR extremum at that moves from at to large values as is lowered.

In [4, 43], the flavor potential was parametrized as in order to satisfy the first requirement (a). This is apparently the simplest Ansatz which works, is motivated by string theory and we will restrict to this form here. More general Ansätze could also be considered, for example quartic terms in the tachyon [56]. In V-QCD it is, however, more essential to include the -dependence in (and possibly in ), as discussed in subsection 2.4. This is also natural in order to reproduce the “running” of the coupling (and the quark mass) of QCD, (see [4], and the next subsection).

The second requirement (b) is necessary in order for the phase diagram to have the desired structure as is varied. Assuming the parametrization discussed above, , the existence of the extremum of

(4.47)

is guaranteed in the BZ region () if the -dependence of and is matched with the -function of QCD. On the field theory side the extremum is mapped to a (perturbative) IR fixed point. For generic values of the existence of the fixed point is nontrivial, and its existence needs to be studied case by case as the phase diagram may be affected.

The simplest Ansatz for the remaining functions and of the flavor action is to take them to be functions of only: and . Again the -dependence of is necessary to reproduce the running of the quark mass in QCD. As both functions appear as couplings under the square root in the DBI action, it is natural to expect that they have qualitatively similar functional form.

The CP-odd action contains two additional functions and . The form of can be constrained by studying Yang-Mills theory. should go to constant in the UV () and diverge as in the IR () [32]. Further constraints have been discussed recently in [57]. The standard Ansatz is therefore

(4.48)

where the constant can be matched to the topological susceptibility of Yang-Mills, and can be fitted to the spectrum of glueballs (see [34] for details). Notice that the value of also depends on the choice for .

4.1 UV structure

The UV properties of most of the functions have been discussed in detail in [4, 43]. We repeat here the main points for completeness. As it turns out, the functions , , and must be analytic in the UV, . Therefore we may expand them as