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Pion Form Factor in Chiral Limit of Hard-Wall AdS/QCD Model

We develop a formalism to calculate form factor and charge density distribution of pion in the chiral limit using the holographic dual model of QCD with hard-wall cutoff. We introduce two conjugate pion wave functions and present analytic expressions for these functions and for the pion form factor. They allow to relate such observables as the pion decay constant and the pion charge electric radius to the values of chiral condensate and hard-wall cutoff scale. The evolution of the pion form factor to large values of the momentum transfer is discussed, and results are compared to existing experimental data.

In general, the vector sector is less sensitive to the infrared (IR) effects, since this symmetry is not broken in QCD. However, the axial-vector sector appears to be very sensitive to the particular way the chiral symmetry is broken or, in other words, to the bulk content and the shape of the IR wall [9].
In this respect, one of the interesting objects to study in the holographic dual models of QCD is the pion. The properties of the pion were studied in various holographic approaches, (see e.g. Refs. [5,6,8,12,13,14,15,16,19,21,22]). In particular, the approach of Ref. [6] (see also recent papers [16,19,21]) managed to reproduce the (Gell-Mann-Oakes-Renner) relation m 2 π ∼ m q between the quark mass m q and mass of the pion m π and also the g ρππ coupling (the coupling between ρ meson and two pions). In Ref. [8], the solution of the pion wave-function equation was explicitly written for the m q = 0 limit.
In this paper, working in the framework of the model proposed in [6] (hard-wall model), we describe a formalism to calculate the form factor and wave functions (and also the density function) of the pion. Since the fits of Ref. [6] give a very small m q ∼ 2 MeV value for the explicit chiral symmetry breaking parameter m q , we consider only the chiral limit m q = 0 of the hard-wall holographic dual model of two-flavor QCD. Resorting to the chiral limit allows us to utilize one of the main advantages of AdS/QCD -the possibility to work with explicit analytic solutions of the basic equations of motion. Expressing the pion form factor in terms of these solutions, we are able, in particular, to extract and analyze the behavior of the pion electric radius in various regions of the holographic parameters space. On the numerical side, we come to the conclusion that the radius of the pion is smaller than what is known from experiment. However, we suggest that, as in case of the radius of the ρ meson, smoothing the IR wall may increase the pion radius.
In our analysis, we introduce and systematically use two types of holographic wave functions Φ(z) and Ψ(z), which are conjugate to each other and basically similar to the analogous objects introduced in our papers [23,24], where we studied vector mesons.
The paper is organized in the following way. We start with recalling, in Section II, the basics of the hard-wall model and some results obtained in Ref. [6], in particular, the form of the relevant action, the eigenvalue equations for bound states and their solutions. In Section III, we describe a formalism for calculating the pion form factor and express it in terms of the two wave functions mentioned above. In Section IV, we discuss the relation of our AdS/QCD results to experimental data. We express the values of the pion decay constant and the pion charge radius in terms of the fundamental parameters of the theory and study their behavior in different regions of the parametric space. At the end, we study the behavior of the pion form factor at large momentum transfer. Finally, we summarize the paper.

A. Action and Equations of Motion
The standard prescription of the holographic model is that there is a correspondence between the 4D vector and axial-vector currents and the corresponding 5D gauge fields: where t a = σ a /2, (a = 1, 2, 3 and σ a are usual Pauli matrices).
In general, one can write A = A ⊥ + A , where A ⊥ and A are transverse and longitudinal components of the axial-vector field. The spontaneous symmetry breaking causes A to be physical and associated with the Goldstone boson, pion in this case. The longitudinal component may be Then ψ a (x, z) corresponds to the pion field. Physics of the axial-vector and pseudoscalar sectors is described by the action where DX = ∂X − iA L X + iXA R , (A L(R) = V ± A) and X(x, z) = v(z)U (x, z)/2 is taken as a product of the chiral field U (x, z) = exp (2it a π a (x, z)) and the function v(z) = m q z+σz 3 containing the chiral symmetry breaking parameters m q and σ, with m q playing the role of the quark mass and σ that of the quark condensate. Expanding U (x, z) in powers of π a gives the relevant piece of the action This Higgs-like mechanism breaks the axial-vector gauge symmetry by bringing a z-dependent mass term in the A-part of the lagrangian. Varying the action with respect to the transverse part of the axial-vector gauge field A a ⊥µ (x, z) and representing the Fourier image of A a ⊥µ (x, z) asÃ a ⊥µ (p, z) we will get the following equation of motion that determines physics of the axial-vector mesons, like A 1 . The axial-vector bulk-to-boundary propagator A(p, z) is introduced by the relationÃ a ⊥µ (p, z) = A(p, z)A a µ (p). It satisfies Eq. (4) with boundary conditions (b.c.) A(p, 0) = 1 and A ′ (p, z 0 ) = 0. Similarly, variation with respect to the Finally, varying with respect to A z produces The pion wave function is determined from Eqs. (5) and (6) with b.c. ∂ z ψ(z 0 ) = 0, ψ(ǫ) = 0 and π(ǫ) = 0.
Within the framework of the model of Ref. [6], it is possible to derive the Gell-Mann-Oakes-Renner relation m 2 π ∼ m q producing massless pion in the m q = 0 limit. Taking p 2 = m 2 π in Eq. (6) gives A perturbative solution in the form of m 2 π expansion was proposed in Ref [6], with ψ(z) = A(0, z)−1 in the lowest order. Then it was shown that, in the m q → 0 limit, π(z) tends to −θ(z − z 0 ) or, roughly speaking, π = −1 in this limit. Since our goal is to calculate the pion form factor in the chiral limit, this approximation will be sufficient for us.

B. Two-Point Function
The spectrum in the axial-current channel consists of the pseudoscalar pion 0|J α A |π(p) = if π p α and axial-vector mesons 0|J α A |A n (p, s) = F A,n ǫ α n (p, s), where F A,n correspond to the n th axialvector meson decay constant (and we ignored the flavor indexes). Thus, the two-point function for the axial-vector currents has the form: where the meson polarization tensor is given by The representation for the two-point function can be also written as in which the second term on the rhs is explicitly transverse to p.
As noted in Ref. [6], using holographic correspondence one can relate the two-point function to [∂ z A(p, z)/z] z=0 and derive that For large spacelike p 2 , Eq. (4) gives the same solution as in case of vector mesons, and the same asymptotic logarithmic behavior, just as expected from QCD.

C. Pion Wave Functions
The longitudinal component of the axial-vector gauge field was defined as A = ∂ψ. In the chiral limit, when p 2 = m 2 π = 0, we have ∂ z π = 0, and the basic equation for ψ, Eq. (5) can be rewritten as the equation for the function Ψ ≡ ψ − π. In the chiral limit, when π(z) → −1, the value of Ψ(ǫ) tends to 1 as ǫ → 0. This value and the b.c. Ψ ′ (z 0 ) = 0 are the same as those for A(p, z) and, furthermore, Eq. (12) coincides with the p 2 = 0 version of equation (4) for A(p, z). Hence, the solution for Ψ(z) coincides with A(0, z): and we may write In our analysis of ρ-meson wave functions in Refs. [23,24], we emphasized that it makes sense to consider also the conjugate functions Φ(z) ∼ Ψ ′ (z)/z of the corresponding Sturm-Liouville equation. As we observed, they are closer in their structure to the usual quantum mechanical bound state wave functions than the solutions of the original equation. In the pion case, it is convenient to define the Φ function as It vanishes at the IR boundary z = z 0 and, according to Eq. (11), is normalized as at the origin. Note also that using Eq. (12) we can express Ψ as derivative of Φ:

III. EXTRACTING PION ELECTROMAGNETIC FORM FACTOR
A. Three-point function To obtain the pion form factor, we need to consider three-point correlation functions. The correlator should include the external EM current J el µ (0) and currents having nonzero projection onto the pion states, e.g. the axial currents J a 5α (x 1 ), J a † 5β (x 2 ) where p 1 , p 2 are the corresponding momenta, with the momentum transfer carried by the EM source being q = p 2 − p 1 (as usual, we denote q 2 = −Q 2 , Q 2 > 0). The spectral representation for the three-point function is a two-dimensional generalization of Eq. (8) where the first term, longitudinal both with respect to p α 1 and p β 2 contains the pion electromagnetic form factor F π (Q 2 ) (normalized by F π (0) = 1), while other pole terms contain the contributions involving axial-vector mesons and are transverse either with respect to p α 1 or p β 2 , or both. Hence, the pion form factor can be extracted from the three-point function using

B. Trilinear Terms in F 2 Part of Action
To obtain form factor from the holographic model, we need the action of the third order in the fields. There are two types of terms contributing to the pion electromagnetic form factor: |DX| 2 term and F 2 terms. Let us consider first the contribution from F 2 terms. They contain V V V , V AA and AV A interactions and may be written as where out the part of the action which is contributing to the 3-point function J 5α J µ J 5β : Introducing Fourier transforms of fields, we define, as usual, V µ (q, z) =Ṽ µ (q)V(q, z) for the vector field, whereṼ µ (q) is the Fourier transform of the 4-dimensional field V µ (x) and V(q, z) is the bulk-to-boundary propagator satisfying the equation with b.c. V(q, 0) = 1 and ∂ z V(q, z 0 ) = 0. It can be written as the sum involving all the bound states in the q-channel, with M m being the mass of the mth bound state and ψ V m (z) its wave function given by a solution of the basic equation of motion in the vector sector. The projection (21) picks out only the longitudinal part A µ (p, z) of the axial-vector field.
where A a µ (p, z) and ψ a (p, z) are the Fourier transforms of A a µ (x, z) and ψ(x, z), respectively. Furthermore, there is only one particle in the expansion over bound state in this case, namely, the massless pion. Thus, we have A a µ (p, z) =Ã a µ (p) ψ(z) and, therefore, This allows us to rewrite A a µ (p, z) in the form involving the longitudinal projector p α p µ /p 2 and the pion wave function ψ(z), which is the solution of the basic equation (5). Using this representation and making Fourier transform of W 3 gives (notice that the second term in Eq.(23) vanishes for longitudinal axial-vector fields). Varying this functional with respect to sources produces the following 3-point function: where, anticipating the limit p 2 and applying the projection suggested by Eq. (21), we will have where J(Q) is the dynamic factor given by the convolution

C. Dynamic Factor and Wave Functions
The vector bulk-to-boundary propagator J (Q, z) ≡ V(iQ, z) for spacelike momenta, entering into the dynamic factor J(Q), satisfies the equation with b.c. J (Q, 0) = 1 and ∂ z J (Q, z 0 ) = 0. Its explicit form is given by One can easily see that J (0, z) = 1. Combining all the factors, we get Integrating by parts and using equations of motion both for J and ψ gives We need also to add the V ππ contribution from the |DX| 2 term of the AdS action (2). It is generated by and its inclusion changes ψ(ψ − π) into (ψ − π) 2 in Eq. (36). The total result (see also Ref. [6]) may be now conveniently expressed in terms of the Ψ ≡ ψ − π wave function Using equation of motion for Ψ(z), one can see that the expression in square brackets coincides and write the form factor as This representation allows one to easily check the normalization where we took into account that J (0, z) = 1 and Φ(z 0 ) = 0. We can also represent our result for the pion form factor as and interpret the function ρ(z) as the radial distribution density, as it was done in Refs. [23,24].
Note that keeping only the first term in square brackets gives an expression similar to our result [23] for the ρ-meson form factor in terms of the function φ 1 conjugate to the solution of the basic equation of motion. The value [25]), namely, φ 1 (0) = g 5 g ρ . Thus, the pion wave function g 5 f π Φ(z) ≡ φ π (z) is a direct analog of the ρ-meson wave function φ 1 (z). Main difference is that, in the pion case, there is also the second term in the form factor expression. The latter, in fact, is necessary to secure correct normalization of the form factor at Q 2 = 0. In Eq. (38), this term is written in terms of the Ψ(z) wave function, but using Eq. (17) we can rewrite it also in terms of Φ(z) or φ π (z):

A. Structure of Pion Wave Functions
Explicit form of the Ψ wave function follows from the solution of Eq. (12): where α = g 5 σ/3 ≈ 1.481 σ (recall that g 5 = √ 2π, see e.g. Ref. [24]). As a result, Φ(z) is given by This formula, combined with Eq. (16), establishes the relation for f π in terms of the condensate parameter α and the confinement radius z 0 . Since σ appears in the solutions only through α, we will use α in what follows. Note also that α 1/3 ≈ 1.14 σ 1/3 .
Realizing that the equations of motion for the vector sector in this holographic model are not affected by the chiral symmetry-breaking effects expressed through the function v(z), it is natural to set the value of z 0 from the vector sector spectrum, i.e., by the ρ-meson mass. The numerical value of z 0 (call it z ρ 0 ) is then z ρ 0 ≈ 1/323 MeV. As given by Eq. (46), f π looks like a rather complicated function of two scales, z 0 and α. Note, however, that the ratio I 2/3 (a)/I −2/3 (a) is very close to 1 for a > ∼ 2 and practically indistinguishable from 1 for a > ∼ 3. Hence, for sufficiently large values of the confinement radius, z 0 > ∼ 1/α 1/3 , the value of f π is determined by the value of α alone. This limiting value of f π is given by Requiring that f π | z 0 →∞ coincides with the experimental value, f π ≈ 131 MeV, one should take α 1/3 ≈ 420 MeV. For such α, the value of 1/α 1/3 is close to z ρ 0 , i.e., we are in the region αz 3 0 ∼ 1 and we may expect that, even if we use exact formula (46) with z 0 = z ρ 0 , the value of f π would not change much. Indeed, to get f π ≈ 131 MeV from Eq. (46) for 1/z 0 = 323 MeV, we should take α 1/3 ≈ 424 MeV ≡ α 1/3 0 . Thus, in this range of parameters, the value of f π is practically in one-to-one correspondence with the value of α. It is convenient to introduce a dimensionless variable a ≡ αz 3 0 = Then the values α 1/3 0 = 424 MeV and 1/z ρ 0 = 323 MeV correspond to a = 2.26 ≡ a 0 . As one can see from Fig.(1), the dependence of f π is practically flat for a > ∼ 2. The confinement radius z 0 presents a natural scale to measure length, so it makes sense to rewrite the form factor formula (38) as an integral over the dimensionless variable ζ ≡ z/z 0 : where the mass scale α is reflected by the dimensionless parameter a. The factor n(a) takes care of the correct normalization of the form factor. It is given by For small a, it may be approximated by 3 4 a 2 . For large a, using the fact that I 2/3 (a)/I −2/3 (a) is very close to 1 for a > ∼ 2, we may approximate n(a) ≈ 0.637 a 2/3 in this region. In terms of n(a), the pion decay constant can be written as For large a, this gives f π a > ∼ 2 ≈ 0.311 α 1/3 .
For small a, we have The functions ϕ(ζ, a), ψ(ζ, a) are just the Φ and Ψ wave functions written in ζ and a variables.
For a = 0, the limiting forms are ϕ(ζ, 0) = 1 − ζ 4 and ψ(ζ, 0) = 1. As a increases, both functions become more and more narrow (see Fig.2). For density, we have ρ(ζ, a = 0) = 4ζ 2  is generated by the monotonically increasing function n(a). It is interesting to compare the pion density ρ(ζ, 2.26) (taken at the "experimental" value a = 2.26) with the ρ-meson density ρ ρ (ζ) of Ref. [23]. These densities are rather close for ζ > 0.5, but strongly differ for small ζ. In particular, the ρ-meson density is more than two times larger for ζ = 0, which corresponds to the hard-wall model result that g ρ is essentially larger than f π .
and explicit form of the density where ν(a) ≡ I 2/3 (a)/I −2/3 (a), we obtain for the pion charge radius: Hence, for fixed z 0 and small a, when α ≪ 1/z 3 0 , the pion radius is basically determined by the confinement scale z 0 . In particular, r 2 π = 4 3 z 2 0 for α = 0. Numerically, taking z 0 = z ρ 0 ≈ 1/323 MeV = 0.619 fm, we obtain r 2 π = 0.51 fm 2 . This result is very close to the value r 2 ρ C ≈ 0.53 fm 2 that we obtained in the hard-wall model for the ρ-meson electric radius determined in [23] from the slope of the G C (Q 2 ) form factor. However, since G C (Q 2 ) involves kinematic-type terms Q 2 /m 2 ρ , it seems more appropriate to compare F π (Q 2 ) with the F 11 (Q 2 ) form factor (42) given directly by a wave function overlap integral. The slope of F 11 (Q 2 ) is smaller than that of G C (Q 2 ), and the corresponding radius is also smaller: r 2 ρ F = 0.27 fm 2 . Thus, for α = 0, the pion r.m.s. radius is about 1.4 times larger than the ρ-meson size determined by r 2 ρ 1/2 F . With the increase of α, the pion becomes smaller (see Fig.4). The experimental value of 0.45 fm 2 [25] is reached for a ∼ 0.9. However, the corresponding value f π ≈ 80 MeV is too small. If we take a = a 0 = 2.26, then r 2 π = 0.34 fm 2 . Thus, if we insist on using z 0 = z ρ 0 dictated by the hard-wall model calculation of the ρ-meson mass, and the value of α producing the experimental f π (note that then α −2/3 ≈ 0.222 fm 2 ), the pion radius is smaller than the experimental value. In linear units, the difference, in fact, does not look very drastic: just 0.58 fm instead of 0.66 fm. Given that the hard-wall model for confinement is rather crude, the agreement may be considered as encouraging.
Furthermore, one may expect that, in a more realistic softer model of confinement, the size of the pion will be larger. Such an expectation is supported by our soft-wall model calculation of the ρ-meson electric radius, for which we obtained r 2 ρ C =0.66 fm 2 (0.40 fm 2 for r 2 ρ F ), i.e., the result by 0.13 fm 2 larger than in the hard-wall model. If r 2 π would increase by a similar amount, the result will be very close to the quoted experimental value.
To find r 2 π for large a (i.e., when α > ∼ z −3 0 for fixed z 0 , or when z 0 > ∼ α −1/3 for fixed α), we use first the observation that, in the region a > ∼ 2, we may approximate ν(a) ≈ 1. Then the factor in square brackets in Eq. (55) becomes a function of the combination aζ 3 ≡ µ (call it R(µ)), and we can write For a > ∼ 2, the upper limit of integration in this expression may be safely substituted by infinity which gives Using Eq. (47), we can express the coefficient in front of the square bracket in terms of f π : Thus, r 2 π in the a > ∼ 2 region consists of two componens: a fixed term 3/4π 2 f 2 π and a term logarithmically increasing with z 0 . As z 0 → ∞, the pion charge radius becomes infinite, reflecting the fact that the pion in this model is massless. A similar structure in the expression for the pion charge radius was obtained [28] in the Nambu-Jona-Lasinio (NJL) model It also has the logarithmic term ln m 2 π [29,30] resulting in the infinite radius for massless pion and the infrared-finite piece 3/2π 2 f 2 π [31,32]. The latter, however, is twice larger than that in our result (60) and contributes 0.34 fm 2 to r 2 π , with the chiral logarithm term producing the extra 0.11 fm 2 required for agreement with experiment. In our case, the logarithmic term taken for a = a 0 is approximately equal to 3/4π 2 f 2 π , thus almost doubling the outcome value for r 2 π . More precisely, we can write For a = 2.26, the modified logarithmic term gives a very small contribution, and our net result is very close to the value given by the NJL fixed term. Numerically, though, this prediction of the hard-wall AdS/QCD model, as we have seen, is essentially smaller than the experimental value.

C. Form Factor at Large Q 2
In the large-Q 2 limit, the source J (Q, z) is given by its free-field version zQK 1 (Qz) that behaves asymptotically like e −Qz . As a result, only small values z ∼ 1/Q are important in the form factor integral, and the large-Q 2 asymptotic behavior of the form factor is determined by the value of ρ(z) at the origin [2,4,24], namely, Note that the combination 4π 2 f 2 π ≡ s 0 ≈ 0.68 GeV 2 frequently appears in the pion studies. In particular, it is the basic scale of the pion wave function in the local quark-hadron duality model [33,34], where it corresponds to the "pion duality interval". The leading contribution comes entirely from the Φ 2 term of the form factor integral (41) while the Ψ 2 term contribution behaves asymptotically like 1/Q 4 since it is accompanied by extra z 2 factor. Note, however, that it is quite visible in the experimentally interesting region Q 2 < ∼ 10 GeV 2 : it is responsible for more than 20% of the form factor value in this region (moreover, at Q 2 = 0, the Ψ 2 term contributes about 40% into the normalization of the form factor).
From a phenomenological point of view, different AdS/QCD-like models for the pion form factor differ in the shape of the density ρ(ζ) that they produce. If we require that the density ρ(z) equals 2π 2 f 2 π at the origin, the asymptotic behavior is F π (Q 2 ) → s 0 /Q 2 in any such model. For Q 2 = 0, the form factor is normalized to one, so basically the models would differ in how they interpolate between these two limits. In particular, the simplest interpolation is provided by the monopole while our hard-wall calculation gives a curve that goes above F mono π (Q 2 ): the ratio F π (Q 2 )/F mono π (Q 2 ) is larger than 1 for all Q 2 > 0, slowly approaching unity as Q 2 → ∞ (see  In fact, a purely monopole form factor was obtained in our paper [24], where we studied the ρ-meson form factors in the soft-wall holographic model, in which confinement is generated by ∼z 2 oscillator-type potential. It was shown in [24] that the form factor integral in which Φ(z, κ) = √ 2 κ e −z 2 κ 2 /2 is the lowest bound state wave function, and is the bulk-to-boundary propagator of this oscillator-type model, is exactly equal to 1/(1 + Q 2 /(4κ 2 )). The magnitude of the oscillator scale κ was fixed in our paper [24] by the value of the ρ-meson mass: κ = κ ρ ≡ m ρ /2. As a result, the form factor F(Q 2 , κ = m ρ /2) had the ρ-dominance behavior 1/(1 + Q 2 /m 2 ρ ). If we take κ = κ π ≡ πf π ≈ 410 MeV both for Φ(z, κ) and J O (Q, z), the integral (65) gives 1/(1 + Q 2 /s 0 ). The relevant wave function Φ(z, κ π ) has the expected correct normalization Φ(0, κ π ) = √ 2πf π , however, the slope 1/s 0 of 1/(1 + Q 2 /s 0 ) at Q 2 = 0 (corresponding to 0.35 fm 2 for the radius squared) is smaller than that of the experimental pion form factor. Furthermore, Q 2 F mono π (Q 2 ) tends to s 0 ≈ 0.68 GeV 2 for large Q 2 , achieving values about 0.5 GeV 2 for Q 2 ∼ 2 GeV 2 , and thus exceeding by more than 25% the experimental JLab values [26] measured for Q 2 = 1.6 and 2.45 GeV 2 . The authors of Ref. [27] proposed to use Eqs. (65),(66) as an AdS/QCD model for the pion form factor, with κ = 375 MeV chosen so as to fit these high-Q 2 data. However, such a choice underestimates the value of f 2 π by almost 30%. Our opinion is that the AdS/QCD models should describe first the low-energy properties of hadrons, and the basic low-energy characteristics, such as m ρ and f π , should be used to fix the model parameters. On the other hand, if the form factor calculations based on these parameters disagree with the large-Q 2 data, it is quite possible that this is just an indication that one is using the model beyond its applicability limits. Furthermore, as we have seen in the hard-wall model, to correctly describe the pion one needs to include the chiral symmetry breaking effects absent in the vector channel.
As a result, equations for pion wave functions are rather different from those in the ρ-meson case.
Similarly, there are no reasons to expect that, in a soft-wall model, the pion density should have the same shape as the ρ-meson one. Unfortunately, the procedure of bringing in the chiral symmetry breaking effects that was used in the hard-wall model of Ref. [6] faces serious difficulties when applied to the AdS/QCD model [9] with the z 2 soft wall. As discussed in Ref. [9], the solution of the equation for the X field in this model requires that chiral condensate σ and the mass parameter m q are proportional to each other, so that σ cannot be varied independently of m q . Moreover, if one takes the chiral limit m q = 0, the chiral condensate should also vanish. This difficulty may be avoided by switching to more sophisticated recent models (cf. [16,17,18]) in which the chiral condensate is generated dynamically. However, such a consideration goes well beyond the scope of the present paper. Thus, we just resort to an idea that whatever the mechanism is involved, the net practical outcome is a particular shape of the density ρ(z) that eventually determines the pion form factor and other pion characteristics. Below, we give an example of a density ρ mod (z) that is normalized at the origin by the experimental value of f π , i.e., ρ mod (0) = 2κ 2 π , but which is also capable to reproduce the experimental value of the pion charge radius.
Evidently, to increase the radius, we should take a density which is larger for large z than Φ 2 (z, κ = πf π ). Since the overall integral normalization of the density is kept fixed, this can be achieved only by decreasing the density for small z values. Consider a simple ansatz (see Fig.7) with A = 1 − κ 2 ρ /κ 2 π + 2B. It has both the desired value for z = 0 and satisfies the normalization condition ∞ 0 dz zρ mod (z) = 1 .
Integrating it with J O (Q, z) taken at κ = κ ρ produces the model form factor given by the following sum of contributions of the three lowest vector states: The slope of F mod π (Q 2 ) at Q 2 = 0 is given by Taking B = 1/4, one obtains the experimental value 0.45 fm 2 for r 2 π . It is interesting to note that the model density providing this value, has an enhancement for larger values of z (see Fig.7), just like the pion densities in the hard-wall model (see Fig.(3)). Due to a larger slope, F mod π (Q 2 ) decreases faster than the simple monopole interpolation F mono π (Q 2 ) and, as a result, is in better agreement with the data. In fact, it goes very close to Q 2 < ∼ 1 GeV 2 data, but exceeds the values of the JLab Q 2 =1.6 and 2.45 GeV 2 points by roughly 10% and 20%, respectively.
This discrepancy has a general reason. The asymptotic AdS/QCD prediction is On the other hand, JLab experimental points correspond to Q 2 F exp π (Q 2 ) ≈ 0.4 GeV 2 , which is much smaller than the theoretical value quoted above. The pre-asymptotic effects, as we have seen, reduce the discrepancy, but there still remains a sizable gap. As we already stated, such a disagreement may be just a signal that we are reaching a region where AdS/QCD models should not be expected to work. In particular, AdS/QCD models of Refs. [2,4,6,8] describe the pion in terms of an effective field or current, without specifying whether the current is built from spin-1/2 fields, or from scalar fields, etc. For Q 2 above 1 GeV 2 , the quark substructure of the pion may be resolved by the electromagnetic probe (which is a wide-spread belief), and the description of the pion "as a whole" may be insufficient.

V. SUMMARY
In this paper, we studied the pion in the chiral limit of two flavor QCD. To this end, we described a formalism that allows to extract pion form factor within the framework of the holographic dual model of QCD with hard-wall cutoff. Following Ref. [6], we identified the pion with the longitudinal component of the axial-vector gauge field. We defined two (Sturm-Liouville) conjugate wave functions Φ(z) and Ψ(z) that describe the structure of the pion along the 5 th dimension coordinate z. These wave functions provide a very convenient framework to study the holographic physics of the pion. We demonstrated that, just like in the ρ-meson case [23], the pion form factor is given by an integral involving the function ρ(z) that has the meaning of the charge density inside the pion. However, in distinction to the ρ-case, when the density was simply given by |Φ(z)| 2 , the pion density has an additional term proportional to |Ψ(z)| 2 and entering with the z-dependent coefficient reflecting the mechanism of the spontaneous symmetry breaking. Both terms are required for normalization of the form factor at Q 2 = 0.
We found an analytic expression for the pion decay constant in terms of two parameters of the model: σ and z 0 , similar to those used in Ref. [8]. Analyzing the results, we found it convenient to work with two combinations α = g 5 σ/3 and a = αz 3 0 of the basic parameters. In particular, we found a = a 0 = 2.26 for the value of a corresponding to the experimental ρ-meson mass m ρ and pion decay constant f π . The importance of the parameter a is that its magnitude determines the regions, where the pion properties are either governed by the confinement effects or by the effects from the spontaneous chiral symmetry breaking. For example, in the practically important domain a > 2, the pion decay constant is determined primarily by σ, with negligibly tiny corrections due to z 0 value. However, when a < 1 the pion decay constant is proportional to the ratio a/z 0 . Besides, for small a ≪ 1, the radius of the pion is given by r 2 π = 4 3 z 2 0 , i.e., as one may expect, the pion size is completely determined by the confinement radius. On the other hand, for a > 2 the radius is basically determined by 1/σ 1/3 , slowly increasing with z 0 due to the ln a/a 0 correction.
We also found that the pion rms charge radius r 2 π 1/2 ≈ 0.58 fm in the hard-wall model is smaller than that measured experimentally. In a sense, the hard wall at the distance z 0 ≈ 0.62 fm (fixed from the ρ-meson mass), "does not allow" the pion to get larger. So, we argued that if the IR wall is "softened", the size of the pion may be increased by an amount sufficient to accomodate the data.
A straightforward idea is to use the soft-wall model of Ref. [9] and treat the pion in a way similar to what was done in [24] for the ρ-meson case. Unfortunately, there are prohibiting complications with directly introducing the chiral symmetry effects within the AdS/QCD model with the z 2 soft wall. As explained in Ref. [9], the chiral condensate σ in such a model is proportional to the mass parameter m q , so that in the chiral limit the condensate vanishes together with the quark mass.
To illustrate a possible change in the form factor predictions due to the softening of the IR wall, we proposed an ansatz for the pion density function and used the vector current source from the soft-wall model considered in Ref. [24]. We demonstrated that this ansatz is capable to fit the experimental value of the pion charge radius. It also closely follows the data in the Q 2 < 1 GeV 2 region, while still overshoots available data in the Q 2 ∼ 2 GeV 2 region. The basic source of this discrepancy is very general: the asymptotic AdS/QCD prediction for the pion form factor is Q 2 F π (Q 2 ) → 4π 2 f 2 π , and if one takes the experimental value for f π , one obtains Q 2 F π (Q 2 ) → 0.68 GeV 2 , which is much larger than the 0.4 GeV 2 value given by Q 2 ∼ 2 GeV 2 JLab data. For this reason, we argued that the disagreement mentioned above may be a signal that the region Q 2 > ∼ 2 GeV 2 is beyond the applicability region of AdS/QCD models.
Finishing the write-up of this paper, we have learned that the paper [35] addressing the same problem was posted into the arxive. We did not observe, however, essential overlaps with our ideas and results.