Note on relation between bottom-up holographic models and large-Nc QCD

We discuss a derivation of the quadratic in fields part of action of bottom-up holographic models from some general properties of the large-Nc limit in QCD.


Introduction
The physics of hadrons composed of light quarks is highly non-perturbative as these hadrons represent strongly coupled systems. Some time ago an interesting proposal of analytical approach to the strongly coupled gauge theories came from the string theory -the hypothesis of AdS/CFT correspondence [1] (also referred to as gauge/gravity duality or holographic duality). It turns out that a description of certain strongly coupled gauge theories can be given in terms of weakly-coupled higher-dimensional gravitational theories. The holographic ideas have penetrated to many branches of modern theoretical physics. In the hadron spectroscopy, these ideas are challenging since the AdS/CFT correspondence provides a practical recipe for calculation of correlation functions in strongly coupled gauge theories. The calculation is performed via the semiclassical expansion of the action of higher-dimensional dual theories [2]. This can have a direct application to the hadron spectroscopy: The masses of hadron states appear as poles of correlation functions of QCD currents interpolating those states.
The concrete holographic prescription is as follows [2]. All correlation functions of a 4D field theory can be obtained from the generating functional of the connected correlators W 4D [ϕ 0 (x)] that depends on the sources ϕ 0 (x) for the 4D field theory operators. If we know the action S 5D of dual theory, the holographic correspondence postulates the identification By assumption, the 5D dual theory is in the weakly coupled regime. This implies a important consequence: As the first approximation, one may consider only the quadratic terms in S 5D and use the semiclassical analysis. The calculation of n-point correlators consists thus in evaluation of S 5D on a solution of equation of motion and subsequent differentiation n times with respect to boundary values of 5D fields. The bottom-up holographic approach represents an attempt to apply the holographic ideas to QCD. The corresponding models try to describe the non-perturbative dynamics of strong interactions in terms of a putative semiclassical five-dimensional theory. The first versions of bottom-up modelsthe so-called Hard Wall (HW) models [3,4] -had a problem with description of linear Regge and radial spectrum (m n ∼ n instead of the behavior m 2 n ∼ n expected in the phenomenology and on some theoretical grounds). The construction of the Soft Wall (SW) holographic model [5] solved this problem.
There is still no consensus why the bottom-up holographic models turned out to be unexpectedly successful in the phenomenology. This success looks surprising and a bit enigmatic. An inherent property of the holographic duality is the large-N c (planar) limit of gauge theories [6]. It is believed therefore that all AdS/QCD models are models for the planar QCD where mesons represent stable and freely propagating particles (their interactions and decays are suppressed by powers of 1/N c [6]). Within a hypothetical dual theory, the corresponding mass spectrum should emerge in the weak coupling regime, i.e., in the first approximation, it must be determined by a quadratic in fields part of action.
The purpose of this work is to demonstrate explicitly that the quadratic in fields parts of actions of typical bottom-up holographic models follow directly from some general expectations about the behavior of correlation functions in the large-N c limit of QCD. Partly this analysis was done in Ref. [7]. But the derivation in [7] did not take into account that, in general case, one must use rescaled (with respect to the holographic coordinate) fields in order to have finite sources. In the present work, we fix this drawback. As a byproduct, we also demonstrate the importance of "weak gravity" approximation which allows to replace the covariant derivatives by the usual ones and to use a important relation for the 5D mass as a function of spin.

The large-N c QCD as a Kaluza-Klein reduction
We recall some general results from the analysis of QCD in the limit of large number of colors N c [6]. The meson masses scale as m ∼ N 0 c and their full decay widths do as Γ ∼ N −1 c . The decay widths thus disappear in the limit of large N c . Since the meson masses vary slightly with N c , this limit is often useful in the meson spectroscopy. In the extreme case N c → ∞, the mesons are infinitely narrow and non-interacting, in addition, an infinite number of states emerge for each set of quantum numbers. Such an infinite tower of resonance poles saturates completely the two-point correlation function of quark currents with quantum numbers I G (J P C ) corresponding to the given tower, With the help of functional integral formalism, this is described by the following action with source terms, ..µ J , µ i = 0, 1, 2, 3 (the sign convention is (+ − −−)), corresponds to a meson field of spin J, the other quantum numbers are unspecified. The tensor φ J is symmetric, traceless, φ µ µ... = 0 (to provide the irreducible (J/2, J/2) representation of the homogeneous Lorentz group), and satisfy the auxiliary condition ∂ µ φ µ... = 0 (to give the required 2J + 1 physical degrees of freedom). The dots denote additional derivative terms appearing in description of free higher spin fields. The auxiliary conditions can be chosen in such a way that they do not contribute (to be discussed below). The tensor structure in the r.h.s. of representation (2) depends on the structure of derivative terms. The sources O (n) J can be represented as where the constants F for a meson φ (n) J with "polarization" ε J and O J is a common source to which the states φ J (n) are coupled with a "coupling" F (J) n . The representation (4) follows from the requirement to get the standard form for two-point correlators: integrating over the field φ J in the generating functional and differentiating twice with respect to O J at O J = 0 one arrives at the sum over meson poles (2). Our task is to rewrite the expression (3) as a Kaluza-Klein reduction of action of some free 5D theory. We will look for the corresponding 5D actions in the form where , and f 1 (z), f 2 (z) are yet unknown functions of the fifth (space-like) coordinate z. We start from a flat 5D space having boundaries in the fifth coordinate, z min ≤ z ≤ z max . The action (7) contains a z-dependent background which is different for the kinetic and mass terms. We will imply the following condition for the physical components of 5D fields, ϕ z... = 0.
With this condition, no additional quadratic terms appear in description of free higher-spin mesons [5,8,9]. The equation of motion for the action (7) with condition (8) is where we used the usual plane-wave ansatz for the physical 4D particles carrying the 4D momentum q n , After imposing boundary conditions on the fields, the equation (9) represents a classical Sturm-Liouville (SL) problem. Consider the following mathematical problem: Given a spectrum m 2 n,J in (3), find the functions f 1 (z) and f 2 (z) which being inserted in (9) would lead to eigenvalues m 2 n,J . The solution of this "inverse" SL problem constitutes the main intermediate step in rewriting the action (3) with sources as some free 5D field theory.
First we recall briefly the main results of the SL theory. The SL equation Here the function p(z) > 0 has a continuous derivative, the functions q(z) > 0 and ω(z) > 0 are continuous on the finite closed interval [z min , z max ]. The SL problem consists in finding the values of λ for which there exists a nontrivial solution of Eq. (11) satisfying certain boundary conditions. Under the assumptions that p(z) −1 , q(z), and ω(z) are real-valued integrable functions over the interval [z min , z max ], with the boundary conditions of the form where α, β ∈ [0, π) and prime means derivative, the SL theorem states that (i) There is an infinite discrete set of real eigenvalues λ n , n = 0, 1, 2, . . . ; (ii) Up to a normalization constant, there is a unique eigenfunction ϕ n (z) corresponding to each eigenvalue λ n and this eigenfunction has exactly n − 1 zeros in [z min , z max ]; (iii) The normalized eigenfunctions form an orthonormal basis Thus the solutions of the SL problem form a complete set of functions in the interval [z min , z max ] which can be used for expansion of arbitrary (but sufficiently smooth) functions in that interval. Under these conditions, the SL problem (9) has solutions ϕ (J) n (z) which are normalized as zmax z min and form a complete set of functions. The function ϕ J (x, z) in (7) can be expanded in the 4D harmonics, Now we substitute the expansion (16) into the action (7), Integrating by parts and making use of Eq. (9), the second term in the action (17) can be rewritten as (dropping the general factor −φ Integrating over z in the action (17) with the help of (15) and (18) we obtain with ϕ J (x, z) given by (16). The action (19) has the form of (3) if we identify The classical Kaluza-Klein reduction corresponds to O Choosing β = 0 or β = π/2 in the boundary condition (13) we can nullify the term at z = z max in Eq. (20). Note in passing that for the vector mesons the first possibility is actually realized in the SW holographic models [5], where z max → ∞, while the second one -in the HW models [3,4], where z max is the infrared cutoff. Thus,

From large-N c QCD to bottom-up holographic models
The action (7) can be rewritten as a "weakly coupled" 5D gravitational theory of free field ϕ J with the metric that dictates the corresponding metric tensor G M N and the background function f 1 (z), We must impose the covariant rule for contraction of indices, where This can be achieved only if the function f 1 (z) in (23) is replaced by the J-dependent background All this just rearranges the notations but does not change our discussions above. The choice in (22) The eigenfunctions ϕ (J) n (z) in the AdS space behave as z 2 at small z for the twist 2 operators, i.e. when the relation (36) holds [7]. This ensures constant values for F (J) n in (29). The appearance of factor c is due to a freedom to redefine F n → cF n , O → O/c where the parameter c can be fixed by matching to the high energy asymptotics of the corresponding QCD correlators. For the case of vector mesons, our way of getting the relation (29) is alternative (up to a general factor depending on normalization of fields) to the derivation of Ref. [3].
The relation (28) shows that if we want, following the AdS/CFT prescriptions, to interpret the source as a finite boundary value of a 5D field, ϕ J (x, +0) = O J , we must rescale the higher spin (J > 1) fields, In this sense, the physical tensor field is the rescaled oneφ J . This statement was obtained in Ref. [8] using a different approach. Now we see that the holographic prescription for calculation of correlation functions in a 4D theory from a "dual" 5D theory is completely reconstructed on the level of quadratic in fields actions. The higher vertices (describing meson decays and scattering) are suppressed in the large-N c limit and we therefore do not consider them. Some clarifying remarks should be made. Since the AdS space is not flat, the usual derivatives must be replaced by the covariant ones in the case of tensor mesons. However, the holographic duality for strongly coupled 4D theories works only if the dual 5D theories are weakly coupled. This means that we should use the "weak gravity" limit (very large AdS radius) as the first approximation. In this limit, the affine connections are neglected as the AdS space locally is almost flat. This justifies the use of usual derivatives for higher spin mesons in our analysis.
The second remark concerns the standard statement that the spectrum of the HW model depends on the boundary conditions. Indeed, the Dirichlet boundary conditions which have been used since the pioneering papers [3,4] correspond to the choice β = π/2 in (13) and seemingly one could use any other choice for β. There is, however, a subtle point here. With other choices, generally speaking, the boundary contribution at z = z max is not zero. This means the appearance of the second source that would look strange for the standard functional technics 1 .
The last remark concerns the choice of background in the 5D action. It can be shown that if we want to reproduce the linear Regge spectrum and simultaneously the correct analytical structure of the operator product expansion of the two-point correlators, the background is fixed unambiguously -it corresponds to nothing but the SW model to be discussed in the next section. The corresponding analysis is presented in detail in Ref. [7].

Spectrum of the soft-wall model
An attractive feature of the SW holographic models is the Regge spectrum for meson masses [5]. The original derivation of Ref. [5] was based on gauge higher spin fields. The condition (8) was interpreted as a gauge choice. A residual invariance after imposing (8) (that canceled the additional derivative terms) allowed to eliminate the mass term for the rescaled fields (30). Below we reproduce the Regge spectrum of Ref. [5] using the massive higher spin fields and our discussions above.
The action of SW model is a particular case of (7) written in the AdS space, where the dilaton background e az 2 was introduced to get the Regge spectrum. Substituting the rescaled field (30) and lowering all indices we find The corresponding equation of motion for physical modes (10) reads The SL equation (33) has the following discrete spectrum m 2 n,J = 2|a| 2n + 1 + This spectrum depends on the 5D mass m J . The free p-form fields in the AdS 5 space have masses m 2 p R 2 = (∆ − p)(∆ + p − 4). As was shown in Ref. [9], the condition (8) leads to decoupling of kinematical aspects and the symmetric tensors of rank p (describing mesons of spin p) obey the same equation as the p-form fields. In the "weak gravity" approximation, this results in the following 5D mass of higher spin fields propagating in the AdS 5 space [9,10], where ∆ is the canonical (conformal) dimension of a 4D field theory operator dual to a corresponding 5D field on the AdS boundary. The spectrum of the original SW model [5] corresponds to the operators of twist 2, i.e. ∆ = J + 2. Substituting this dimension to (35), we have The spectrum (34) reduces then to the spectrum, This spectrum does not depend on spin if a < 0. For a > 0 one arrives at a remarkably simple Regge spectrum m 2 n,J = 4a(n + J) that coincides with the spectrum of poles of the Veneziano amplitude.
For the purposes of phenomenological applications, it can be interesting to construct an extension of the SW model that leads to a shifted Regge spectrum, m 2 n,J = 4a(n + J + b), where b is a phenomenological parameter regulating the intercept of Regge and radial trajectories. The matter is that this form of light meson spectrum (with b = 0) seems to be supported by the experimental data [11]. The relevant holographic extension was constructed in Ref. [12] for the vector case. The method of Ref. [12] can be generalized to higher spins. We will not display the derivation in detail and just give the final answer that can be checked by a direct calculation: The dilaton background e az 2 must be replaced by where U is the Tricomi hypergeometric function.

In conclusion
In essence, we tried to demonstrate that the bottom-up holographic approach to QCD represents nothing but a five-dimensional language for expressing some general expectations from the large-N c limit in QCD. This interpretation is valid only for holographic actions quadratic in fields.
The action of SW model is purely phenomenological and constructed with the aim of getting the linear Regge spectrum. It is not known which 5D dynamical model results in the background e az 2 as a solution of 5D Einstein equations. An "intermediate" dynamical model that leads to such a background could look as follows, where the action is written in the AdS 5 space and L represents a Lagrangian density for the meson sector. The dilaton coupling in the form e φ L is typical in the string theory. If the 5D mass squared m 2 of the dilaton field φ takes the minimal value permitted in the AdS 5 space, m 2 R 2 = −4 (the Breitenlohner-Freedman stability bound [13]) and one neglects backreaction from L, then the equation of motion for φ will have a solution φ = az 2 . This solution leads to the z-dependent background of SW model. One may speculate that the dilaton field φ in (39) mimics the non-perturbative gluodynamics. In the large-N c limit, the gluon sector dominates. For instance, the probability of creation of glueballs scales as O(N 2 c ) while that of mesons scales as O(N c ) [6]. We do not see an abundant production of glueballs in the real world just because we cannot create experimentally the corresponding gluon currents. But theoretically the scaling O(N 2 c ) of gluon sector justifies neglecting backreaction from L in the first approximation. In any case, the simplest SW model should be viewed as an approximation in which a dynamical dilaton field is replaced by a static dilaton background dictated by the phenomenology.
The use of the bottom-up models is much broader than just modelling the hadron spectra -it includes description of the spontaneous chiral symmetry breaking in QCD, hadron formfactors, QCD phase diagram, etc. The corresponding applications usually contain certain extensions of simple bottom-up models that go beyond the large-N c limit or have no clear connections with the planar QCD. Establishing these connections is an open problem.