Strong couplings and form factors of charmed mesons in holographic QCD
Alfonso Ballon-Bayona Gastao Krein Carlisson Miller



















    ß⟨q̅ q ⟩
            
            
            
            
            
            
            
            
            
            aballonb@ift.unesp.brgkrein@ift.unesp.brmiller@ift.unesp.brInstituto de Física Teórica, Universidade Estadual
              Paulista,  Rua Dr. Bento Teobaldo Ferraz, 271 - Bloco II,
              01140-070 São Paulo, SP, Brazil
            We extend the two-flavor hard-wall holographic model of Erlich, Katz, Son and Stephanov [Phys. Rev. Lett. 95, 261602 (2005)] to four flavors to incorporate strange and charm quarks. The model incorporates chiral
            and flavor symmetry breaking and provides a reasonable description of masses and weak decay constants
            of a variety of scalar, pseudoscalar, vector and axial-vector strange and charmed mesons. In particular, we examine flavor
            symmetry breaking in the strong couplings of the ρ meson to the charmed D and D^* mesons.
            We also compute electromagnetic form factors of the π,  ρ, K, K^*, D and D^* mesons. We compare our results for the D and D^* mesons
            with lattice QCD data and other nonperturbative approaches.
            Strong couplings and form factors of charmed mesons in holographic QCD
                Carlisson Miller
                
            ======================================================================
            
            
            
            
            § INTRODUCTION
            
            
            There is considerable current theoretical and experimental interest in the study of the interactions
            of charmed hadrons with light hadrons and atomic nuclei <cit.>. There is special interest in the properties of D mesons in nuclear matter <cit.>,
            mainly in connection with D-mesic nuclei <cit.>, J/Ψ and η_c binding to nuclei <cit.>, and ND molecules <cit.>. D-mesons are also of interest in the context of the
            so-called X,Y,Z exotic hadrons, which have galvanized the field of hadron spectroscopy since the discovery in 2003
            of the charmed hadron X(3872) by the Belle collaboration <cit.>. They are  exotic because they
            do not fit the conventional quark-model pattern of either quark-antiquark mesons or three-quark baryons. Most of
            the X,Y,Z hadrons have masses close to open-flavor thresholds and decay into hadrons containing charm (or bottom)
            quarks. Presently there is no clear theoretical understanding of the new hadrons, despite of the huge literature
            that has accumulated over the last decade. In the coming years, existing and forthcoming experiments will produce
            numerous new, and very likely surprising results—Ref. <cit.> is
            a very recent review on exotic hadrons, with an extensive list of references on theory and experiment.
            The PANDA collaboration <cit.>, in particular, at the forthcoming FAIR facility has an
            extensive program <cit.> aiming at the investigation of charmed hadrons and
            their interactions with ordinary matter.
            
            A major difficulty in the theoretical treatment of in-medium interactions of charmed hadrons is the lack
            of experimental information on the interactions in free space. For example, almost all knowledge on the DN
            interaction comes from calculations based on effective Lagrangians that are extensions of light-flavor
            chiral Lagrangians using SU(4) flavor symmetry <cit.> and heavy quark
            symmetry <cit.>. The Lagrangians involve coupling constants, like
            g_ρDD, g_ωDD, g_ρD^∗ D and g_ρD^∗ D^∗, whose values are taken
            from SU(4) flavor and heavy-quark symmetry relations. For instance, SU(4) symmetry relates the couplings of
            the ρ to the pseudoscalar mesons π, K and D, namely g_ρDD = g_KKρ = g_ρππ/2.
            If in addition to SU(4) flavor symmetry, heavy-quark spin symmetry is invoked, one has g_ρ DD =
            g_ρ D^∗ D = g_ρ D^∗ D^∗ = g_π D^∗ D to leading order in the charm quark
            mass <cit.>. The coupling g_ρππ is constrained by experimental
            data; the studies of the DN interaction in
            Refs <cit.> utilized such a SU(4) relation, taking
            g_ρππ = 6.0, which is the value used in a large body of work conducted within the Jülich
            model <cit.> for light-flavor hadrons. This value of g_ρππ
            implies through SU(4) symmetry g_ρ DD = 3, which is not very much different from predictions based on
            the vector meson dominance (VMD) model: g_DDρ = 2.52-2.8 <cit.>. Moreover, to maintain
            unitarity in calculations of scattering phase shifts and cross sections, lowest-order Born diagrams need to be
            iterated with the use of a scattering equation, like the Lippmann-Schwinger equation, and phenomenological
            form factors are required to control ultraviolet divergences. Form factors involve cutoff parameters that
            also are subject to flavor dependence. Again, due to the lack of experimental information, they
            are also poorly constrained.
            
            Flavor symmetry is strongly broken at the level of the QCD Lagrangian due to the widely different values of the
            quark masses; while in the light quark sector one has good SU(2) symmetry, m_u ≃ m_d, thereby e.g.
            g_ρDD = g_ωDD (up to a phase), in the heavy-flavor sector SU(3) and SU(4) symmetries are
            badly broken: m_c ≫ m_s ≫ m_u. Given the importance of effective Lagrangians in the study of a
            great variety of phenomena involving D-mesons, in the present examine their properties in a holographic model
            of QCD. We extend the holographic QCD model of  Refs. <cit.> to the case of N_f=4
            and investigate the implications of the widely different values of the quark masses on the effective three-meson
            couplings g_ρ D D and g_ρ D^* D^* and the electromagnetic form factors of the D and D^* mesons.
            The parameters of the model are the quark masses and condensates as well as the mass gap scale.  Using experimental
            data for a selected set of meson masses to fix the model parameters, allows us to predict not only
            the strong couplings and electromagnetic form factors mentioned above but also many other observables not
            studied before with a holographic model.
            
            The works in Refs. <cit.> pioneered in the modeling of low energy QCD by incorporating
            features of dynamical chiral symmetry breaking in holographic QCD. They correctly identify the five dimensional
            gauge fields dual to the left and right currents
            associated with chiral symmetry as well as the five dimensional scalar field dual to the chiral condensate.
            The extension proposed in Ref. <cit.> incorporated the strange quark and was able to identify
            the appearance of scalar modes associated with flavor symmetry breaking. In the present work, by extending the
            model of Refs. <cit.> to the case N_f=4, we are able to investigate the
            consequences of the dramatically different values of the quark masses on the phenomenology of charmed mesons.
            Moreover, by combining the formalism of Kaluza-Klein expansions and the AdS/CFT dictionary, we are able to directly
            extract the leptonic decay constants of mesons and find an expansion for the flavor currents that relate flavor
            symmetry breaking to the appearance of scalar modes. That relation bears a strong analogy with the generalized
            PCAC (partially conserved axial current relation) <cit.> that relates dynamical
            chiral symmetry breaking to the appearance of the pion and its resonances <cit.>.
            
            In the model of  Refs. <cit.>, dynamical chiral symmetry breaking becomes manifest
            when considering fluctuations of the five dimensional gauge fields associated with the axial and vectorial sector.
            While the kinetic terms of the axial sector acquire a mass, signalizing chiral symmetry breaking, the vector
            sector remains massless. In our framework, it turns out that the vector sector also acquires a mass signalizing
            the breaking of flavor symmetry. The Kaluza-Klein decomposition of these fields  allows us to obtain effective
            kinetic Lagrangians for the mesons from the five dimensional kinetic terms, with masses and decay constants
            obtained in terms of the wave functions representing the Kaluza-Klein modes. Moreover, expanding the five
            dimensional action to cubic order in the fluctuations and performing again a Kaluza-Klein decomposition allows
            us to obtain effective Lagrangians describing the three-meson interactions, with strong couplings given in terms
            of integrals involving the wave functions of the corresponding Kaluza-Klein modes.
            
            It turns out that the symmetry breaking pattern in the strong couplings differs somewhat from previous studies
            in the literature. Calculations employing QCD sum rules found SU(4) symmetry breaking in three-hadron couplings
            that vary within the range of 7% to 70%<cit.>. In Ref. <cit.>, using a model constrained by the Dyson-Schwinger equations of QCD, it was found that the relation g_ρ DD =
            g_ρππ/2 is strongly violated at the level of 300% or more. In a recent follow up of that study
            within the same framework, Ref. <cit.> finds that couplings between D-, D^∗-mesons
            and π-, ρ-mesons can differ by almost an order-of-magnitude, and that the corresponding form factors
            also exhibit different momentum dependences. Our results calculations are more in line with calculations
            using the  ^3 P_0 quark-pair creation model in the nonrelativistic quark model <cit.>.
            
            The organization of this paper is as follows. In Sec. <ref> we describe how chiral and flavor
            symmetry breaking is realized in our model. Then in Sec. <ref> we describe the five dimensional
            field equations and the AdS/CFT dictionary for the flavor and axial currents. In Sec. <ref> we
            describe the formalism of Kaluza-Klein expansions and obtain effective kinetic Lagrangians for the mesons.
            In Sec. <ref> we use the prescription of our previous studies in Ref.  <cit.>
            for the leptonic decay constants and obtain relations describing flavor symmetry breaking and chiral symmetry
            breaking in terms of scalar and pseudoscalar modes respectively. In Sec. <ref> we obtain
            effective Lagrangians describing three-meson interactions with the holographic prescription for the strong
            couplings. Finally, in Sec. <ref> we fit the model parameters and present our numerical results
            for many observables, including the strong couplings g_ρ DD and g_ρ D^* D^* as well as
            the electromagnetic form factors of the D and D^* mesons. We compare the latter against lattice QCD
            data obtained in Ref. <cit.>. Section <ref> presents our conclusions.
            
            
            
            
            
            § CHIRAL SYMMETRY AND FLAVOR SYMMETRY IN HOLOGRAPHIC QCD
            
            
            Chiral symmetry SU(N_f)_L × SU(N_f)_R for N_f flavors holds in the massless limit of QCD
            and is described in terms of the left and right currents
            
            
            J_L/R^μ , a = q̅_L/Rγ^μ T^a q_L/R  ,
            
            where T^a, a=1, … N^2_f-1 are the generators of the SU(N_f) group,
            and q_L/R = 1/2 (1±γ_5) q, with q being the quark Dirac field. The SU(4) generators
            T^a are normalized by the trace condition Tr ( T^a T^b ) = 1/2  δ^ab, satisfying the
            Lie algebra [ T^a , T^b  ] = i f^abc T^c. The generators T^a are related to the
            Gell-Mann matrices λ^a by T^a = 1/2  λ^a. Chiral symmetry is broken by the presence
            of the operator q̅ q = q̅_R q_L + q̅_L q_R. This breaking can be
            explicit, when it appears in the QCD Lagragian associated with the nonzero quark masses, or dynamically, when
            it acquires a vacuum expectation value, giving rise to a condensate ⟨q̅ q⟩ in limit of
            zero quark masses.
            
            In the case N_f=2, dynamical chiral symmetry breaking goes as SU(2)_L × SU(2)_R → SU(2)_V,
            where SU(2)_V is an exact vector symmetry and the broken symmetry occurs in the axial sector. This is described
            in terms of the vector and axial currents J^μ,a_V/A= J^μ,a_R ± J^μ , a_L. The symmetry
            associated with the vector sector J^μ,a_V is known as isospin symmetry. When N_f >2, both chiral and
            flavor symmetries are broken by the quark masses. We will describe how chiral and flavor symmetry breaking are
            implemented in a holographic model for N_f=4.
            
            In the pioneering work of Refs. <cit.> and <cit.>, a simple holographic realization
            of chiral symmetry breaking (CSB) was proposed. They considered the simplest background in holographic QCD,
            known as the hard wall model <cit.>, consisting in a slice of anti-de-Sitter spacetime:
            
            
            ds^2 = 1/z^2 ( η_μν dx^μ dx^ν - dz^2    )    , 
            with 0 < z  ≤ z_0. The parameter z_0 determines an infrared (IR) scale at which conformal symmetry
            is broken. The action proposed in Ref. <cit.> includes N_f gauge fields L_m and R_m,
            corresponding to the left and right flavor currents J_L/R^μ , a, and a bifundamental field X dual
            to the operator q̅_R q_L.  The action can be written as
            
            
            S    =   ∫ d^5 x √(|g|) Tr{ (D^m X)^† (D_m X)  + 3 |X|^2    -   1/4 g_5^2 ( L^mn L_mn + R^mn R_mn )
            }  ,  
            where D_m X = ∂_m X - i L_m X + i X R_m is the covariant derivative of the bifundamental field X,
            and
            
            
            L_mn   =   ∂_m L_n - ∂_n L_m - i  [ L_m , L_n  ], 
            
            R_mn   =   ∂_m R_n - ∂_n R_m - i  [ R_m , R_n  ],
            
            are non-Abelian field strengths. The 5-d squared mass of the field X is fixed to  m^2 = -3, to match
            with the conformal dimension Δ=3 of the dual operator q̅_R q_L. The model of Ref. <cit.>
            focused on N_f=2 and worked in the limit of exact flavor symmetry. In Ref. <cit.>,
            Abidin and Carlson extended the model to N_f=3, to incorporate the strange-quark sector. In the present
            paper we further extend that model to N_f=4, with the aim of making predictions for charmed mesons. In our
            approach we use a Kaluza-Klein expansion for the 5-d fields in order to find a 4-d effective action for the
            mesons. This approach allows us to find directly the meson weak decay constants, couplings and expansions
            for the vector and axial currents. We find in particular a relation for the vector
            current describing flavor symmetry breaking (FSB).
            
            We start with the classical background that describes chiral symmetry breaking:
            
            
            L^0_m =     R^0_m = 0   ,    2 X_0 = ζ M z + Σ/ζ z^3 ,
            
            where M is the quark-mass matrix, M= diag(m_u , m_u , m_s , m_c), and Σ is the matrix of the
            quark condensates, Σ= diag(σ_u ,  σ_u , σ_s , σ_c). The parameter ζ=√(N_c)/2π  is introduced to have consistency with the counting rules of large-N_c  QCD—for details, see Ref. <cit.>. Note that we are assuming SU(2) isospin symmetry in the light-quark sector, i.e. m_d = m_u and σ_d = σ_u, which is a very good approximate symmetry in QCD.
            
            For the strange and charm quarks we will fit their masses m_s and m_c to the physical masses for the mesons. Note, however, that the model should not be valid for arbitrarily large quark masses. The reason is that, from the
            string theory perspective, the action in equation (<ref>) is expected to arise from a small perturbation
            of N_f coincident space-filling flavor branes. Specifically, the mass term M appearing in
            Eq. (<ref>) acts as a small source for the operator q̅_R q_L, responsible for the
            breaking of the chiral  and flavor symmetries. A holographic description of quarks with very large masses requires
            the inclusion of long open strings and two sets of flavor branes distinguishing the heavy quarks from the
            light quarks (see e.g. <cit.>). In that framework, the string length is proportional to the
            quark mass and each set of flavor branes will carry a set of fields describing the dynamics of light and
            heavy mesons respectively. In this work we will show that the model described by the action in
            equation (<ref>) is still a very good approximation for the dynamics of light and
            heavy-light charmed mesons, the reason being that the internal structure in both cases is governed by
            essentially the same nonperturbative physics, that occurs at the scale Λ_ QCD <cit.>.
            In heavy-heavy mesons, on the other hand, the internal dynamics is governed by short-distance
            physics. For recent holographic studies of mesons involving heavy quarks see
            Refs. <cit.>.
            
            To investigate the consequences of chiral and flavor symmetry breaking it is convenient to rewrite the fluctuations
            of left and right gauge fields in terms of vector and axial fields , i.e. L_m = V_m + A_m and
            R_m = V_m - A_m.  The bifundamental field X can be decomposed as
            
            
            X = e^iπ  X_0   e^iπ,
            
            where X_0 is the classical part and π contains the fluctuations. The fields V_m, A_m and π
            can be expanded as V_m^a T^a, A_m^a T^a and π^a T^a respectively.
            
            It is important to remark that organizing the heavy-light D mesons together with the light pions and kaons in a 15-plet π^a T^a of fluctuation  
            fields does not imply, automatically, that the heavy-light D mesons are being approximated by Nambu-Goldstone bosons. The reason is that the explicit breaking of chiral symmetry, driven by the 
            heavy charm quark, is large and by no means its effects are neglected in the model. In the same way, 
            the fact the D mesons appear in the same multiplet of the SU(4) flavor group does not mean that flavor symmetry is exact; it is explicitly broken by the widely different values of the quark masses. The 
            main advantage of using such a SU(4) representation with explicit symmetry-breaking terms is that it 
            allows us to make contact with the four dimensional effective field theories describing the interactions
            of light and heavy-light mesons commonly used in phenomenological applications. This not only extends 
            the work of  <cit.> but also leads to quantitative predictions for the strong couplings that can 
            be tested against experiment or lattice QCD data, which is our main objective in the present paper. 
            An alternative approach to describe the heavy-light mesons is to make contact with a particularly interesting class of four dimensional models that treat
            the light mesons as in the present paper, and treat heavy mesons by invoking heavy-quark symmetry. The Lagrangian
            in the heavy sector is written as an expansion in inverse powers of the heavy quark mass; Refs. <cit.> are examples of such models. In holography the heavy quarks are realized in terms of long open strings, as described above in this section. For recent progress in the heavy quark approach to heavy-light mesons within holographic QCD see <cit.>.
            
            Expanding the action in Eq. (<ref>) up to cubic order in the fields V_m^a, A_m^a and π^a,
            we find
            
            
            S = S^(2) + S^(3) + …  ,
            
            where
            
            
            S^(2)   =   ∫ d^5 x √(|g|){ - 1/4 g_5^2 v^mn_a v_mn^a + 1/2(M^a_V)^2
            V^m_a V_m^a - 1/4 g_5^2 a^mn_a a_mn^a
            +  M_A^a b/2 (∂^m π^a - A^m,a) (∂_m π_b - A_m,b) }
            S^(3)   =   ∫ d^5 x √(|g|){ - 1/2g_5^2 f^abc  v^mn_a  ( V_m^b V_n^c + A_m^b A_n^c  )
            - 1/g_5^2 f^abc a^mn_a V_m^b A_n^c
             - (M_V^b)^2/2 f^abc   (∂_m π^a - 2 A_m^a) V^m,bπ^c     +    M_A^ae f^ebc   (∂_m π^a - A_m^a) V^m,bπ^c }  ,  
            and we have defined the Abelian field strengths v_mn^a = ∂_m V_n^a - ∂_n V_m^a
            and a_mn^a = ∂_m A_n^a - ∂_n A_m^a. In the kinetic term S^(2), the vector and axial
            symmetry breaking is dictated by the mass terms M_V^a and M̃_A^a b,
            defined by the traces 2  Tr ( [T^a , X_0] [ T^b, X_0] )  = - (M_V^a)^2 δ^ab and 2  Tr ( { T^a , X_0 }{ T^b , X_0 } ) =  M_A^ab. Note, however, that the axial sector in S^(2) is invariant
            under the gauge transformation
            
            
            A^a_m→ A^a_m-∂_mλ_A^a   ,   π^a→π^a - λ_A^a  . 
            
            Using Eq. (<ref>) we find the following nonzero values for M_V:
            
                  
            (M_V^a)^2 =   1/4 (v_s - v_u)^2     for   a= (4,5,6,7)   ,       
            (M_V^a)^2 =   1/4 (v_c - v_u)^2     for   a= (9,10,11,12)   ,       (M_V^a)^2 =   1/4 (v_c - v_s)^2     for   a= (13,14)   , 
            and the nonzero values for M_A:
            
                  M_A^a,a = v_u^2             for   a= (1,2,3)   ,        M_A^a,a = 1/4 (v_s + v_u)^2   for   a= (4,5,6,7)   ,        M_A^a,a =  1/4 (v_c + v_u)^2    for   a= (9,10,11,12)   ,        M_A^a,a = 1/4 (v_c + v_s)^2    for   a= (13,14)   ,       M_A^8,8 = 1/3 (v_u^2 + 2 v_s^2 )   ,        M_A^15,15 = 1/12 (2 v_u^2 + v_s^2 + 9 v_c^2 )   ,         M_A^8,15 = M_A^15,8 = 1/3 √(2) (  v_u^2  -  v_s^2)   .  
            In Eqs. (<ref>) and (<ref>) we have defined
            
            
            v_q (z) = ζ m_q z + 1/ζσ_q z^3     ,     q=(u,s,c)  .
            
            
            In the interesting case where all the masses and condensates are equal we have that (M_V^a)^2=0
            and the SU(4) flavor symmetry is preserved. In this paper we consider quark masses and condensates that lead to a realistic
            spectrum for the mesons so that we could explore the consequences of SU(4)  flavor symmetry breaking. The kinetic term in Eq. (<ref>) allows us to  extract the meson spectrum and decay
            constants whereas the action in Eq. (<ref>) leads to nontrivial predictions for three-meson couplings,
            including the heavy-light charmed mesons D and D^*.
            
            Note that the mass term for the vectorial sector (M_V^a)^2 is zero not only for a=(1,2,3), corresponding to the light  SU(2) sector but also for a=(8,15), which implies that flavor symmetry has not been broken in the sector describing the dynamics of the ω' and ψ mesons. This is one clear example of heavy-heavy mesons (mesons composed by a heavy quark-antiquark pair), where we actually expect some corrections to appear in (<ref>) describing flavor symmetry breaking. Those terms would arise from the dynamics of long open strings dual to heavy quarks, as explained above in this section.
            
            
            
            
            
            § FIELD EQUATIONS AND DUAL CURRENTS
            
            
            Writing the kinetic action in Eq. (<ref>) as S^(2) = ∫ d^5 x   L^(2), its variation
            takes the form δ S^(2) =
            
            δ S^(2)_ Bulk + δ S^(2)_ Bdy where
            δ S^(2)_ Bulk   =   ∫ d^5 x   [  ( ∂ L^(2)/∂ V_ℓ^a - ∂_m P_V , a ^m ℓ )  δ V_ℓ^a
                +    ( ∂ L^(2)/∂ A_ℓ^a
            - ∂_m P_A , a ^m ℓ )  δ A_ℓ^a    +    ( ∂ L^(2)/∂π^a
            - ∂_m P_π , a^m  )  δπ^a ]   ,
            δ S^(2)_ Bdy = ∫ d^5 x  ∂_m  ( P_V , a^m ℓδ V_ℓ^a + P_A , a^m ℓ δ A_ℓ^a + P_π , a^m  δπ^a  )   ,       
            and
            
            P_V , a ^m ℓ   :=   ∂ L^(2)/∂ (∂_m V_ℓ^a)  = - 1/g_5^2√(|g|)   v^m ℓ_a    , 
            P_A , a ^m ℓ   :=   ∂ L^(2)/∂ (∂_m A_ℓ^a)  = - 1/g_5^2√(|g|)   a^m ℓ_a   , 
            P_π , a^m    :=   ∂ L^(2)/∂ ( ∂_m π^a )  = M_A^ab√(|g|)  (∂^m π_b - A^m_b )   ,
            
            are conjugate momenta to the 5-d fields V_m^a, A_m^a and π^a respectively.
            
            
            
            
            
            
            
            
            The bulk term in Eq. (<ref>) leads to the field equations
                  ∂_m  [ √(|g|) v^mn_a  ] + g_5^2 (M_V^a)^2 √(|g|)   V^n_a = 0   ,       ∂_m  [ √(|g|) a^mn_a  ] - g_5^2 M_A^ab√(|g|)  (∂^n π_b - A^n_b ) = 0   ,       ∂_m  [ M_A^ab√(|g|)  (∂^m π_b - A^m_b )  ] = 0  . 
            
            
            
            
            
            
            Imposing the boundary conditions  ∂_z V_μ^a |_z=z_0 = ∂_z A_μ^a |_z=z_0 = 0 and ∂_z π^a |_z=z_0 = A_z |_z=z_0 = V_z |_z=z_0 =  0 the boundary term (<ref>) reduces to
            
            
            
            
            
            δ S^(2)_ Bdy   =    - ∫ d^4 x    [ ⟨ J^μ̂_V,a⟩ ( δ V_μ̂^a )_z = ϵ
            +  ⟨ J^μ̂_A,a⟩ (δ A_μ̂^a)_z = ϵ   +   ⟨ J_π,a⟩ (δπ^a )_z = ϵ ] ,
            
            where we find the dual 4-d currents
            ⟨ J^μ̂_V,a (x) ⟩   =     P_V , a^z μ|_z = ϵ
            = - 1/g_5^2 ( √(|g|) v^z μ_a  )_z = ϵ ,
            
            
            ⟨ J^μ̂_A,a (x) ⟩   =     P_A , a^z μ|_z = ϵ
            = - 1/g_5^2 ( √(|g|) a^z μ_a  )_z = ϵ  ,
            
            
            ⟨ J_π,a (x) ⟩   =   
             P_π , a^z |_z = ϵ
            =    [ √(|g|) M_A^ab ( ∂^z π_b - A^z_b  )  ]_z = ϵ   =   ∂_μ̂⟨ J^μ̂_A,a (x) ⟩ .
            
            Note that we distinguish the vector Minkowski indices μ̂ from the AdS indices μ.
            The results in Eqs. (<ref>), (<ref>) and (<ref>) define the holographic prescription for
            expectation values of the 4-d vector, axial and pion current operators.
            Note from Eqs. (<ref>) and (<ref>) that ∂_μ̂⟨ J^μ̂_V,a (x) ⟩≠ 0 when (M_V^a)^2 ≠ 0, i.e. the vector current is not conserved for those cases. Similarly,
            from Eqs. (<ref>) and (<ref>), one sees that  ∂_μ̂⟨ J^μ̂_A,a (x) ⟩≠ 0 for any a (because M_A^ab≠ 0), i.e. the axial current is never conserved.
            
            
            
            
            
            § THE 4-D EFFECTIVE ACTION
            
            
            After decomposing the vector and axial fields into their (z,μ) components and evaluating the metric
            in Eq. (<ref>), the kinetic action in Eq. (<ref>) takes the form
            
            S^(2)   =    S^(2)_V + S^(2)_A  ,
            
            where
            
            S^(2)_V   =   ∫ d^4 x ∫d z/z{  - 1/4 g_5^2 [ (v_μ̂ν̂^a)^2 - 2 (v_z μ̂^a)^2  ]    +   (M_V^a)^2/2 z^2 [ (V_μ̂^a)^2 - (V_z^a)^2  ] }  , 
            and
            
            S^(2)_A   =   ∫ d^4 x ∫d z/z{ - 1/4 g_5^2 [ (a_μ̂ν̂^a)^2 - 2 (a_z μ̂^a)^2  ]    +   M_A^ab/2z^2 [ (∂^μ̂π^a - A^μ̂, a ) (∂_μ̂π^b - A_μ̂^b)    -    (∂_z π^a - A_z^a ) (∂_z π^b - A_z^b)  ]
            }. 
            
            
            The vector and axial sectors admit a decomposition in irreducible representations of the Lorentz group.
            For the vector sector we find
            
            V_μ̂^a    =    V_μ̂^⊥, a + ∂_μ̂  (ϕ̃^a - π̃^a ),
            
            
            V_z^a    =    - ∂_z π̃^a, 
            
            where ∂_μ̂ V_μ̂^⊥, a = 0. The 5-d field V_μ̂^⊥, a describes
            an infinite tower of 4-d massive spin 1 fields, i.e. the vector mesons, whereas the 5-d fields ϕ̃^a
            and π̃^a describe an infinite tower of massive scalar fields, i.e. scalar mesons associated
            with flavor symmetry breaking (FSB).
            
            On the other hand, the gauge symmetry in Eq. (<ref>) allows us to decompose the axial sector
            as
            
            
            A_μ̂^a    →    A_μ̂^⊥ , a   ,   
            A_z^a → - ∂_z ϕ^ a   , π^a    →   π^a - ϕ^ a   , 
            where ∂_μ̂ A_μ̂^⊥, a = 0. This time the 5-d field A_μ̂^⊥ , a
            will describe an infinite tower of 4-d massive axial spin 1 fields, i.e. the axial vector mesons. The 5-d fields
            ϕ^a and π^a will describe an infinite tower of 4-d pseudoscalar fields, i.e the pions associated with
            chiral symmetry breaking (CSB).
            
            Using Eqs. (<ref>), (<ref>) and (<ref>)  the actions in Eqs. (<ref>) and (<ref>) take
            the form
            
            S^(2)_V   =   ∫ d^4 x ∫dz/z{ - 1/4g_5^2 [ (v_μ̂ν^⊥,a)^2 - 2 (∂_z V_μ̂^⊥,a)^2
                -    2 (∂_z ∂_μ̂ϕ̃^a )^2  ] + (M_V^a)^2/2 z^2 [ ( V_μ̂^⊥, a )^2 + (∂_μ̂π̃^a - ∂_μ̂ϕ̃^a )^2    -    (∂_z π̃^a)^2  ]
             + ∂_μ̂ (…) } , 
            S^(2)_A   =   ∫ d^4 x ∫dz/z - 1/4g_5^2 [ (a_μ̂ν^⊥,a)^2
            - 2 (∂_z A_μ̂^⊥,a)^2    -    2 (∂_z ∂_μ̂ϕ^a )^2  ]
            + M_A^ab/2 z^2 [ A_⊥^μ̂,a A_μ̂^⊥, b   +    ( ∂^μ̂π^a - ∂^μ̂ϕ^a) ( ∂_μ̂π^b - ∂_μ̂ϕ^b) - (∂_z π^a) (∂_z π^b)  ]
               +   ∂_μ̂ (…) }, 
            where the terms in (…) are surface terms that vanish after choosing periodic boundary conditions for the fields.
            
            The actions in Eqs. (<ref>) and (<ref>) are in a suitable form to perform a Kaluza-Klein expansion for the
            5-d fields. Before performing this expansion note that the nondiagonal mass term M_A^8,15 induces a
            mixing in the axial sector for meson states with flavor indices  a=(8,15). In this paper we are mainly
            interested in the axial sector states with a=(9,..,14) and a=(1,2,3), corresponding to the heavy-light
            charmed mesons and the usual light mesons. Then from now on we will consider for the axial sector only those states
            where a ≠ (8,15). The axial sector states corresponding to a=(8,15) have an interesting physical
            interpretation, e.g η - η_c mixing for the pseudoscalar sector, and deserve a further study that
            will be pursued in a future project.
            
            The 5-d fields in the vector sector admit a Kaluza-Klein expansion of the form
            
            V_μ̂^⊥, a (x,z)    =    g_5    v^a,n(z) V̂_μ̂^a,n (x)      , 
            π̃^a(x,z)    =    g_5  π̃^a,n(z) π̂_V^a,n(x)   , 
            ϕ̃^a(x,z)    =    g_5  ϕ̃^a,n(z) π̂_V^a,n(x)
              ,   ,
            
            where a sum from n=0 to n=∞ is implicit.  A similar decomposition holds for the 5-d fields in the axial sector:
            
            
            A_μ̂^⊥, a (x,z)    =    g_5    a^a,n(z) Â_μ̂^a,n (x)    , 
            π^a(x,z)    =    g_5  π^a,n(z) π̂^a,n(x)    , 
            ϕ^a(x,z)    =    g_5  ϕ^a,n(z) π̂^ a,n(x)  .  
            Using these expansions the actions in Eqs. (<ref>) and (<ref>) factorize into z integrals and x
            integrals and we find S^(2)_V = ∫ d^4 x  L_V and S^(2)_A = ∫ d^4 x  L_A,
            with the vector and axial 4-d Lagrangians given by
            
             L_V   =     - 1/4Δ_V^a, nmv̂_μ̂ν̂^a,nv̂^μ̂ν̂_a,m
            +1/2 M_V^a,nmV̂_μ̂^a,nV̂^μ̂_a,m   +   1/2Δ_π_V^a,nm (∂_μ̂π̂_V^a,n) (∂^μ̂π_V^a,m)
            - 1/2 M_π_V^a,nmπ̂_V^a,nπ̂_V^a,m  ,        L_A   =     - 1/4Δ_A^a, nmâ_μ̂ν̂^a,nâ^μ̂ν̂_a,m
            +1/2  M_A^a,nmÂ_μ̂^a,nÂ^μ̂_a,m   +   1/2Δ_π^a,nm(∂_μ̂π̂^ a,n)(∂^μ̂π̂^a,m)
            -1/2  M_π^a,nmπ̂^a,nπ̂^a,m  ,       
            with coefficients defined by the z integrals
            Δ_V^a,nm   =   ∫dz/z v^a,n(z) v^a,m(z) , 
            M_V^a,nm   =   ∫dz/z{ [ ∂_z v^a,n(z) ] [ ∂_z v^a,m(z) ]    +   β^a_V(z) v^a,n(z) v^a,m(z) } , Δ_π_V^a,nm   =   ∫dz/z{ [ ∂_z ϕ̃^a,n (z) ]  [ ∂_z ϕ̃^a,m (z) ]    +   β^a_V(z) [ π̃^a,n(z) - ϕ̃^a,n(z) ] [ π̃^a,m(z) - ϕ̃^a,m(z) ]
            } , 
            M_π_V^a,nm   =   ∫dz/zβ_V^a(z) [ ∂_z π̃^a,n ] [ ∂_z π̃^a,m ],
            
            for the vector sector and
            
            Δ_A^a,nm   =   ∫dz/z a^a,n(z) a^a,m(z)    , 
            M_A^a ,nm   =   ∫dz/z{ [ ∂_z a^a,n(z) ] [ ∂_z a^a ,m(z) ]    +   β^a_A(z) a^a,n(z) a^a ,m(z) }  , Δ_π^a,nm   =   ∫dz/z{ [ ∂_z  ϕ^a,n (z) ]  [ ∂_z  ϕ^a,m (z) ]     +   β^a_A(z) [  π^a,n(z) -  ϕ^a,n(z) ] [ π^a,m(z) -  ϕ^a,m(z) ]  }  , 
            M_π^a,nm   =   ∫dz/zβ_A^a(z) [ ∂_z  π^a,n ] [ ∂_z  π^ a,m ],
            
            for the axial sector. Here we have defined
            
            β_V^a := g_5^2/z^2  (M_V^a)^2     ,   β_A^a := g_5^2/z^2 M_A^aa .
            
            
            In order to obtain standard kinetic terms in Eqs. (<ref>) and (<ref>) we impose the following
            conditions:
            
                  Δ_V^a, n m = Δ_π_V^a, nm =
            Δ_A^a , nm = Δ_π^a , n m = δ^nm  ,        M_V^a , nm = m_V^a,n^2 δ^nm   ,   
            M_π_V^a, nm = m_π_V^a,n^2 δ^nm  ,        M_A^a , nm = m_A^a,n^2 δ^nm   ,   
            M_π^a, nm = m_π^a,n^2 δ^nm  .       
            The Lagrangians in Eqs. (<ref>) and (<ref>) then reduce to
             L_V    =     - 1/4v̂_μ̂ν̂^a,nv̂^μ̂ν̂_a,n
            +1/2 m_V^a,n^2 V̂_μ̂^a,nV̂^μ̂_a,n   +   1/2  (∂_μ̂π̂_V^a,n) (∂^μ̂π_V^a,n)
            - 1/2 m_π_V^a,n^2 π̂_V^a,nπ̂_V^a,n  ,   L_A    =     - 1/4â_μ̂ν̂^a,nâ^μ̂ν̂_a,n
            +1/2  m_A^a,n^2 Â_μ̂^a,nÂ^μ̂_a,n   +   1/2  (∂_μ̂π̂^a,n)(∂^μ̂π̂^a,n)
            -1/2  m_π^a,n^2 π̂^a,nπ̂^a,n  . 
            The conditions for the Δ coefficients are normalization rules for the corresponding wave
             functions. The conditions for the masses are equivalent to the conditions for the Δ coefficients
             if we impose the following equations :
                   [ - ∂_z  ( 1/z∂_z    )
            +  1/zβ_V^a(z)  ]  v^a,n(z)  = m_V^a,n^2/z
            v^a,n(z)   ,       β_V^a(z)/z[π̃^a,n(z) - ϕ̃^a,n(z) ] =
            - ∂_z  [1/z∂_z ϕ̃^a,n(z)  ]   ,       β_V^a(z)  ∂_z π̃^a,n(z) = m_π_V^a,n^2 ∂_z
            ϕ̃^a,n(z)
              , 
            for the vector sector and
                   [ - ∂_z  ( 1/z∂_z    )
            +  1/zβ_A^a(z)  ]  a^a,n(z) = m_A^ a,n^2/z a^a,n(z)
               ,       β_A^ a(z)/z[π^a,n(z) - ϕ^a,n(z) ] =
            - ∂_z  [1/z∂_z ϕ^a,n(z)  ]   ,       β_A^a(z)  ∂_z π^a,n(z) = m_π^a,n^2 ∂_z
            ϕ^a,n(z)   , 
            for the axial sector.
            
            We finish this section writing the SU(4) pseudoscalar and vector  meson matrices π̂ and V̂
            in terms of the charged states
            
            π̂   =   π̂^a T^a = 1/√(2) (π^0/√(2)  + η/√(6)  + η_c/√(12)   π^+     K^+   D̅^0  
            π^-    -π^0/√(2)  + η/√(6)  + η_c/√(12)     K^0       D^-
            
               K^-   K̅^0      - √(2/3)η + η_c/√(12)     D_s^-
            
               D^0      D^+     D_s^+    - 3/√(12)η_c
              ) , 
            
            V̂   =   V̂^a T^a  = 1/√(2) ( ρ^0/√(2) + ω'/√(6) + ψ/√(12)   ρ^+     K^⋆ +   D̅^⋆ 0
            ρ^-    -ρ^0/√(2)  + ω'/√(6)  + ψ/√(12)     K^⋆ 0     D^⋆ -
            
               K^⋆ -   K̅^⋆ 0    - √(2/3)ω' + ψ/√(12)     D_s^⋆ -
            
               D^⋆ 0     D^⋆ +     D_s^⋆ +    - 3/√(12)ψ ), 
            where in the last equation we have omitted the index μ for simplicity.
            
            
            § DECAY CONSTANTS, CSB AND FSB
            
            
            As observed in <cit.>, the simplest method for extracting the leptonic decay
            constants is to replace the fields in the r.h.s of the dual currents prescription,  Eqs. (<ref>)-(<ref>),
            by their  Kaluza-Klein expansions in Eqs. (<ref>)-(<ref>). We find:
            
            ⟨ J^μ̂_V,a  (x) ⟩   =    [  1/g_5 z∂_z v^a,n(z)   ]_z=ϵV̂^μ̂_a,n(x)     +    [  1/g_5 z∂_z ϕ̃^a,n(z)   ]_z=ϵ∂^μ̂π̂^a,n_V(x)   ,
            
            
            ⟨ J^μ̂_A,  a  (x) ⟩   =    [  1/g_5 z ∂_z a^ a ,n(z)    ]_z=ϵÂ^μ̂_ a,n(x)    +    [ 1/g_5 z∂_z ϕ^ a , n(z)  ]_z=ϵ∂^μ̂π̂^a ,n(x) , 
            
            ∂_μ̂⟨ J^μ̂_A, a (x) ⟩   =   ⟨ J_Π , a  (x) ⟩   =      -   [  β^a_A (z) /g_5 z∂_z π^ a, n(z)
             ]_z=ϵπ̂^a , n(x) ,
            
            where a sum from n=0 to n=∞ is implicit. In the expansions (<ref>)-(<ref>)
            the 4-d fields V̂^μ̂_a,n(x) , A^μ̂_a,n(x) ,  π̂^a,n_V(x) and π̂^a̅ , n(x) are on-shell. From these expansions we find the holographic prescription for leptonic decay
            constants:
            
            g_V^a,n   =    [  1/g_5 z∂_z v^a,n(z)   ]_z=ϵ  , 
            
            f_π_V^a,n   =    -  [  1/g_5 z∂_z ϕ̃^a,n(z)   ]_z=ϵ  , 
            
            g_A^a , n   =    [  1/g_5 z ∂_z a^a ,n(z)    ]_z=ϵ  , 
            
            f_π^ a , n   =    -  [ 1/g_5 z∂_z ϕ^ a , n(z)  ]_z=ϵ  . 
            Taking the divergence of Eqs. (<ref>) and (<ref>) we find
            ∂_μ̂⟨ J^μ̂_V,a  (x) ⟩ =   f_π_V^a,n m_π_V^a,n^2 π̂^a,n_V(x)    , 
            and
            ∂_μ̂⟨ J^μ̂_A,a  (x) ⟩ =   f_π^a,n m_π^a,n^2 π̂^a,n(x)
              , 
            where a sum from n=0 to n=∞ is implicit. Equation (<ref>) is a generalization of the partially
            conserved axial current relation (PCAC), which encodes the effect of chiral symmetry breaking (CSB) in the
            current algebra. Equation (<ref>) encodes the effect of flavor symmetry breaking (FSB) in the vector current. Interestingly, the scalar mesons π̂^a,n_V(x) of FSB  and the pions  π̂^a,n(x) of CSB
            appear in a similar way in Eqs. (<ref>) and (<ref>), respectively.
            
            
            
            § COUPLING CONSTANTS AND FORM FACTORS
            
            
            The three-point interactions are described by the 5-d action in Eq. (<ref>). After decomposing the fields
            into their (z,μ) components and evaluating the metric in Eq. (<ref>), the action takes the
            form
            
            
            S_3 = S_VVV + S_VAA + S_V A  π + S_V  ππ  ,
            
            where
            
            S_VVV   =    - 1/2g_5^2 f^abc∫ d^4 x ∫dz/z [ v^μ̂ν̂_a V_μ̂^b V_ν̂^c    -    2 v_z μ̂^a V_z^b V^μ̂_c   ]   , 
            S_VAA   =   1/2 g_5^2  f^abc∫ d^4 x ∫dz/z [ v^μ̂ν̂_a A_μ̂^b A_ν̂^c - 2 v_z μ̂^a A_z^b A_c^μ̂   -    2 a^μ̂ν̂_a V_μ̂^b A_ν̂^c + 2 a_z μ̂^a (V_z^b A^μ̂_c
            - V^μ̂_b A_z^c ) ]   , 
            S_VA  π   =   1/g_5^2 f^abc∫ d^4 x ∫dz/z [ β_V^b(z)  - β_A^a (z)   ]     ×    [ - A_z^a V_z^b + A_μ̂^a V^μ̂_b ] π^c   , 
            S_V  ππ   =   1/2g_5^2 f^abc∫ d^4 x ∫dz/z [ -  β_V^b(z) + 2 β_A^a(z)   ]    ×    [ - (∂_z π^a) V_z^b + (∂_μ̂π^a ) V^μ̂_b  ] π^ c   . 
            
            
            Here we are interested in the following 4-d triple couplings
            
            S_V̂V̂V̂ =  g__V̂^a , ℓV̂^b, mV̂^c,n∫ d^4 x  V̂^μ̂_a ,ℓ v̂_μ̂ν̂^b, m   V^ν̂_c,n  , 
            S_V̂π̂π̂ =  g__π̂^a,ℓV̂^b,mπ̂^c,n∫ d^4 x   (∂_μ̂π̂^a , ℓ ) V̂^μ̂_b,mπ̂^c,n , 
            where a sum over a,b,c as well as ℓ, m, n is implicit.
            
            Using (<ref>)-(<ref>)  in (<ref>), (<ref>) and (<ref>) as well as the KK expansions (<ref>), (<ref>) and (<ref>)  we find that
            
             g__V̂^a , ℓV̂^b, mV̂^c,n   =   g_5/2 f^abc∫dz/z v^a,ℓ (z) v^b,m(z) v^c,n(z)   ,             g__π̂^ a,ℓV̂^b,mπ̂^c,n = g_5/2 f^a b c∫dz/z{ 2 (∂_z ϕ^ a , ℓ) v^b,m (∂_z ϕ^ c ,n)    +    [ - β_V^b (z) + 2 β_A^ a (z)  ] (π^a,ℓ- ϕ^a,ℓ) v^b,m ( π^c,n - ϕ^c,n) } .       
            
            In order to compare our results with chiral theory models we rewrite the 3-point interactions
            in Eqs. (<ref>) and (<ref>) as
            
            
            S_V̂V̂V̂ =  2 f^abcg̅__V̂^a , ℓV̂^b, mV̂^c,n∫ d^4 x  V̂^μ̂_a ,ℓ   (∂_μ̂V̂_ν̂^b, m)    V^ν̂_c,n  ,
                  
            S_V̂π̂π̂   =    f^abcg̅__π̂^a,ℓV̂^b,mπ̂^c,n∫ d^4 x  V̂^μ̂_b,m [ (∂_μ̂π̂^a , ℓ ) π̂^c,n   -    (∂_μ̂π̂^c , n ) π̂^a,ℓ ]  ,  
            where
            g̅__V̂^a , ℓV̂^b, mV̂^c,n = g_5/2∫dz/z v^a,ℓ (z) v^b,m(z) v^c,n(z) , 
            and
            g̅__π̂^a,ℓV̂^b,mπ̂^c,n   =   g_5/8∫dz/zv^b,m{ 4 (∂_z ϕ^a , ℓ)  (∂_z ϕ^c ,n)    +    [ - 2 β_V^b (z) + 2 (β_A^a(z) +β_A^c(z))  ]    ×    (π^a,ℓ- ϕ^a,ℓ)  ( π^c,n - ϕ^c,n) } . 
            To arrive at Eqs. (<ref>) and (<ref>), we have integrated by parts the actions
            in Eqs. (<ref>) and (<ref>) and also used the transversality of the vector mesons
            (∂_μ̂V̂^μ̂_a ,ℓ=0). Note that the coupling in Eq. (<ref>)
            is symmetric when interchanging the pion flavor indices a and c, as required by crossing symmetry.
            
            We are interested in describing strong couplings involving  charmed mesons D and D^*, strange mesons K  and K^* as well as light mesons π and ρ. Then in (<ref>) and (<ref>)    we select only a=(1,..,7) and a=(9,..,12) for the pseudoscalar mesons π^a, whereas for the vector mesons we pick a=(1..,7), a=(9,..,12) and a=(8,15). The other states are taken to zero.  The reason we include a=(8,15) in the vectorial sector is because it contributes to the electromagnetic form factors of the D and D^* as shown below in this section.  Using also the results in (<ref>)-(<ref>) and evaluating the SU(4) structure constants f^abc we arrive at the effective Lagrangians
            
             L_V ππ   =    L_π D^* D +  L_ρ D D +  L_ω' D D +  L_ψ D D +  L_π K^* K  +  L_ρ K K +   L_ω' K K +   L_ρππ,
            
             L_VVV =  L_ρ D^* D^* +   L_ω' D^* D^* +   L_ψ D^* D^*
            +  L_ρ K^* K^* +  L_ω' K^* K^* +   L_ρρρ ,
            
            where
            
             L_π D^* D   =    i√(2)  g_π  D^* D[D_μ^*+(D̅^0∂^μπ ^-)
            + D_μ^*-( π^+∂^μ  D^0 )
            + D_μ^*0(D^-∂^μπ ^+)
            + D̅_μ^*0( π^-∂^μ   D^+)] 
               +     i g_π D^* D [ D_μ^*+( π^0 ∂^μ D^-)
            + D_μ^*-( D^+∂^μπ^0 )
            + D_μ^*0(D̅^0 ∂^μπ^0 )
            + D̅_μ^*0(  π ^0 ∂^μ D^0 )], 
             L_ρ D D   =    i√(2)  g_ρ D D [
            ρ_μ^+(D^0 ∂^μD^-) + ρ_μ^-( D^+∂^μD̅^0 ) ]
            + ig_ρ DD [ ρ_μ^0(D^-∂^μD^+)
            + ρ_μ^0(D^0∂^μD̅^0 )
            ], 
             L_ω' D D   =   i/√(3)  g_ω' D D [ ω'_μ ( D^+∂^μ D^- )
             + ω'_μ ( D^0 ∂^μD̅^0  )   ], 
             L_ψ D D   =   
             i √(8/3)  g_ψ D D [ ψ_μ ( D^+∂^μ D^- ) + ψ_μ ( D^0 ∂^μD̅^0  )  ], 
             L_ρ D^* D^*   =    i√(2)  g_ρ D^* D^* [ D_μ^*+ (  ρ^-_ν∂^μD̅^ν_*0 )
            +  D_μ^*- ( D_ν^*0∂^μρ_+^ν )
            +  D_μ^*0 ( ρ^+_ν∂^μ D^ν_*- )
            + D̅_μ^*0 ( D_ν^*+∂^μρ_-^ν )
            
               +   ρ_μ^+ ( D^*-_ν∂^μ  D^ν_*0 )
            + ρ_μ^- ( D̅_ν^*0∂^μ D_*+^ν )     ]
            + i g_ρ D^*D^* [  D_μ^*+ ( D_ν^*-∂^μρ_0^ν )  + D_μ^*- ( ρ^0_ν∂^μ  D^ν_*+ )
            
               +     D_μ^*0 ( ρ^0_ν∂^μD̅^ν_*0 )
            + D̅_μ^*0 ( D_ν^*0∂^μρ _0^ν )
            +  ρ_μ^0  ( D_ν^*+∂^μ D_*-^ν )
            +   ρ_μ^0  ( D̅_ν^*0∂^μ  D_*0^ν )
              ], 
             L_ω' D^* D^*   =    - i/√(3) g_ω' D^* D^* [  D_μ^*+ ( D_ν^*-∂^μω'^ν ) + D_μ^*- ( ω'^ν∂^μ  D_ν^*+ ) + D_μ^*0 ( D̅_ν^*0∂^μω'^ν )
             + D̅_μ^*0 ( ω'^ν∂^μ   D_ν^*0 )
            
               +   ω'_μ (D_ν^*+∂^μ D_*-^ν )
            + ω'_μ (D_ν^*0∂^μD̅_*0^ν )
               ],
             L_ψ D^* D^*   =    - i √(8/3) g_ψ D^* D^* [  D_μ^*+ ( D_ν^*-∂^μψ^ν ) + D_μ^*- ( ψ^ν∂^μ D_ν^*+ )
            +  D_μ^*0 ( D̅_ν^*0∂^μψ^ν )
            + D̅_μ^*0 ( ψ^ν∂^μ D_ν^*0 )
            
               +   ψ_μ (D_ν^*+∂^μ D_*-^ν )
            + ψ_μ (D_ν^*0∂^μD̅_*0^ν )
               ],
             L_π K^* K   =    i √(2)  g_π K^* K [ K_μ^*+ ( π^-∂^μK̅^0  ) + K_μ^*- ( K^0 ∂^μπ^+ )
            + K_μ^*0 ( π^+∂^μ K^- )
            + K̅_μ^*0 ( K^+∂^μπ^- )  ] 
               +    i   g_π K^* K [ K_μ^*+ ( π^0 ∂^μ K^- )
            + K_μ^*- ( K^+∂^μπ^0  )
            + K_μ^*0 ( K̅^0∂^μπ^0  )
            + K̅_μ^*0 ( π^0 ∂^μ K^0  )  ] ,
             L_ρ K K   =    i √(2)  g_ρ K K [ ρ_μ^+ ( K^-∂^μ K^0  ) + ρ_μ^- ( K̅^0 ∂^μ K^+ )  ]
            + i g_ρ K K [ ρ_μ^0  ( K^-∂^μ K^+ ) + ρ_μ^0  ( K^0 ∂^μK̅^0  )  ],
             L_ω' K K   =    i √(3)  g_ω' K K [ ω'_μ ( K^-∂^μ K^+ )
            +ω'_μ ( K̅^0 ∂^μ K^0  )  ] ,
            
             L_ρ K^* K^*   =    i √(2) g_ρ K^* K^* [ K_μ^*+ ( K̅_ν^*0∂^μρ_-^ν ) + K_μ^*- ( ρ_ν^+∂^μ K_ν^*0 )
            + K_μ^*0 ( K_ν^*-∂^μρ_+^ν )
            + K̅_μ^*0 ( ρ_ν^-∂^μ  K_*+^ν )
            
               +   ρ_μ^+ ( K_ν^*0∂^μ K_*-^ν )
            + ρ_μ^- ( K_ν^*+∂^μK̅_*0^ν )  ]
            + i  g_ρ K^* K^* [ K_μ^*+ ( K_ν^*-∂^μρ_0^ν ) + K_μ^*- ( ρ_ν^0 ∂^μ K_*+^ν ) 
               +    K_μ^*0 ( ρ_ν^0 ∂^μK̅_*0^ν ) + K̅_μ^*0 ( K_ν^*0∂^μρ_0^ν )
            +ρ_μ^0  ( K_ν^*+∂^μ K_*-^ν ) + ρ_μ^0  ( K̅_ν^*0∂^μ K_*0^ν )  ] ,
            
             L_ω' K^* K^*   =    i √(3)  g_ω' K^* K^* [ K_μ^*+ ( K_ν^*-∂^μω'^ν )  + K_μ^*- ( ω'_ν∂^μ K_*+^ν ) + K_μ^*0 ( K̅_ν^*0∂^μω'^ν ) + K̅_μ^*0 (  ω'_ν∂^μ K_*0^ν )
            
               +   ω'_μ ( K_ν^*+∂^μ K_*-^ν )
            + ω'_μ ( K_ν^*0∂^μK̅_*0^ν )    ],
             L_ρππ   =    i g_ρππ [ρ_μ^+(π^0 ∂^μπ^-)
            +ρ_μ^-(π^+∂^μπ^0 )
            + ρ_μ^0(π^-∂^μπ ^+)
             ], 
            
             L_ρρρ   =    i g_ρρρ[ ρ_μ^+ ( ρ_ν^-∂^μρ_0^ν )  + ρ_μ^- ( ρ_ν^0 ∂^μρ_+^ν )
            + ρ_μ^0  ( ρ _ν^+∂^μρ_-^ν )
              ] .
            
            In the above, the couplings are given by
            
            
            g__π D^* D   =   g̅__π̂^aV̂^bπ̂^c ,    a=(1,2,3)  ,  (b,c)=(9,..,12)  , 
            g__ρ D D   =   g̅__π̂^aV̂^bπ̂^c ,    (a,c)=(9,..,12)  ,   b =(1,2,3)  , 
            g__ω' D D   =   g̅__π̂^aV̂^bπ̂^c ,    (a,c)=(9,..,12)  ,   b =8  , 
            g__ψ D D   =   g̅__π̂^aV̂^bπ̂^c ,    (a,c)=(9,..,12)  ,   b =15  , 
            g__ρ D^* D^*   =   g̅__V̂^a V̂^b V̂^c ,     a=(1,2,3)   ,   (b,c)=(9,..,12)  ,  
            g__ω' D^* D^*   =   g̅__V̂^a V̂^b V̂^c ,     a=8   ,   (b,c)=(9,..,12)  , 
            g__ψ D^* D^*   =   g̅__V̂^a V̂^b V̂^c ,     a=15   ,   (b,c)=(9,..,12)  , 
            g__π K^* K   =   g̅__π̂^aV̂^bπ̂^c ,    a=(1,2,3)  ,  (b,c)=(4,..,7)  , 
            g__ρ K K   =   g̅__π̂^aV̂^bπ̂^c ,    (a,c)=(4,..,7)  ,   b =(1,2,3)  , 
            g__ω' K K   =   g̅__π̂^aV̂^bπ̂^c ,    (a,c)=(4,..,7)  ,   b =8  , 
            g__ρ K^* K^*   =   g̅__V̂^a V̂^b V̂^c ,     a=(1,2,3)   ,   (b,c)=(4,..,7)  ,  
            g__ω' K^* K^*   =   g̅__V̂^a V̂^b V̂^c ,     a=8   ,   (b,c)=(4,..,7)  , 
            g__ρππ   =    2 g̅__π̂^aV̂^bπ̂^c ,     (a,b,c) =(1,2,3)  , 
            g__ρρρ   =     2 g̅__V̂^a V̂^b  V̂^c ,     (a,b,c) =(1,2,3)  . 
            We have used the double arrow derivative  f∂^μg:= f (∂^μ g)
            - (∂^μ f) g and for simplicity we have omitted the indices ℓ , m, n that distinguish the fundamental states from the corresponding resonances. The Lagrangians in Eqs. (<ref>) and (<ref>) are typically used in phenomenology
            of charmed mesons—see e.g. Ref.<cit.>.
            
            
            In the limit where the quark masses and condensates are equal, flavor symmetry is recovered and the
            couplings satisfy the relations
            
            
            g_π D^* D = g_ρ D D  = g_ω' D D = g_ψ D D = 
            g_π K^* K = g_ρ K K = g_ω' K K  = 1/2 g_ρππ =: g/4 ,
            
            g_ρ D^* D^* = g_ω' D^* D^* =  g_ψ D^* D^* = 
            g_ρ K^* K^* = g_ω' K^* K^* =   1/2 g_ρρρ  := g̃/4 .
            
            In this case all the couplings can be obtained from the interaction terms i g  Tr (∂^μπ [ π, V_μ]) and i g̃ Tr (∂^μ V^ν [ V_μ , V_ν ] )—see e.g.
            Ref. <cit.>.
            
            
            
             §.§ Electromagnetic form factors
            
            
            The effective Lagrangian (<ref>) describes the interaction between a vector meson and two pseudoscalar mesons.
            If the vector meson is off-shell and the pseudoscalar mesons are on-shell we can use (<ref>) to investigate the electromagnetic (EM) form factors of pseudoscalar mesons. Similarly, using the effective Lagrangian (<ref>) and taking one of the vector mesons off-shell we can investigate the EM form factors of vector mesons.
            The (elastic) EM form factors of pseudoscalar mesons appear in the decomposition of the EM current as
            ⟨π^a (p+q) | J^μ_EM (0) |π^a (p) ⟩   =    (2p + q)^μ F_π^a(q^2).
            
            Similarly, the (elastic) EM form factors of vector mesons appear as the Lorentz scalars in the EM current decomposition <cit.>
                ⟨ V^a (p+q),ϵ' | J^μ_EM (0) | V^a (p),ϵ⟩   = -(ϵ' ·ϵ)
                (2p + q)^μ F_V^a^1(q^2)    + [ϵ'^μ(ϵ· q)-ϵ^μ(ϵ' · q)]
                [F_V^a^1(q^2) + F_V^a^2(q^2)]    + 1/M_V^a^2(q ·ϵ')(q ·ϵ)(2p + q)^μ F_V^a^3(q^2)  .
            
            Linear combinations of the form factors in (<ref>) define the so called electric, magnetic and quadrupole form factors:
            
            F_V^a^E   =    F_V^a^1 + q^2/6 M_V^a^2[F_V^a^2 - (1-q^2/4M_V^a^2) F_V^a^3 ]   , 
            F_V^a^M   =    F_V^a^1 + F_V^a^2   , 
            F_V^a^Q   =    - F_V^a^2 + (1-q^2/4M_V^a^2) F_V^a^3  . 
            In the absence of baryonic number the EM current is obtained from a linear combination of flavor currents, i.e.
            
            J^μ_EM (x) = ∑_a=(3,8,15) c_a J^μ_a (x)  , 
            where the coefficients c_a can vary depending on the quarks that will be considered in the EM current.
            When considering the EM form factors of the heavy-light D and D^* charmed mesons the strange quark does not participate in the process and we can define the EM
            current as
            
            J^μ_EM = 2/3u̅γ^μ u - 1/3d̅γ^μ d + 2/3c̅γ^μ c .
            
            Then the EM current can be decomposed as (<ref>) with coefficients  c_3=1, c_8=7/(3√(3)) and c_15=- 8/(3√(6)), up to the strangeness current which do not contribute when evaluating the current at the external states. On the other hand, when evaluating the EM form factors for the strange K and K^* as well as the light mesons π and ρ we define the EM current as
            
            J^μ_EM = 2/3u̅γ^μ u - 1/3d̅γ^μ d - 1/3s̅γ^μ s ,
            
            which admits the decomposition (<ref>) for the coefficients c_3=1, c_8 = 1/√(3) and c_15=0.
            
            As explained in the previous section, each flavor current admits a decomposition in terms of vector mesons.
            This implies from (<ref>) that the photon decays into ρ^0,n,  ω'^n and ψ^n mesons. This is is a holographic realization of generalized vector meson dominance (GVMD) <cit.>, in that also the resonances are included and not only the fundamental states as VMD.
            
            For the pion and ρ meson only the states ρ^0,n contribute to the EM form factors. In the case of strange mesons K and K^* the states ρ^0,n and ω'^n contribute to the EM form factors whereas in the case of the charmed D and D^* mesons we have contributions from ρ^0,n, ω'^n and ψ^n. In our model it turns out that the states ω'^n and ψ^n are identical to the states  ρ^0,n as well as their couplings to external states. Although this implies an unrealistic spectrum for those mesons, its contribution to the EM form factors is not only required by consistency but also leads to reasonable results consistent either with experimental data or lattice QCD data, as we will show below.
            
            Using the Feynman rules for the vector meson propagator in  (<ref>) and the triple vertex (<ref>) as well as the EM current decomposition (<ref>), with the appropriate coefficients, we extract the (elastic) EM form factors for the pion, kaon  and D meson :
            
            F_π(Q^2)   =   ∑_ng_ρ^n g_ρ^n ππ/m_ρ^n^2 + Q^2  ,
            
            
            F_K(Q^2)    =   ∑_n [g_ρ^n g_ρ^n K K/m_ρ^n^2 + Q^2 +  g_ω'^n g_ω'^n K K/m_ω'^n^2 + Q^2 ]   =    2 ∑_ng_ρ^n g_ρ^n K K/m_ρ^n^2 + Q^2  ,
            
            
            F_D(Q^2)    =   ∑_n [g_ρ^n g_ρ^n D D/m_ρ^n^2 + Q^2 - 7/9g_ω'^n g_ω'^n D D/m_ω'^n^2 + Q^2   +   16/9g_ψ^n g_ψ^n D D/m_ψ^n^2 + Q^2 ] = 2 ∑_ng_ρ^n g_ρ^n D D/m_ρ^n^2 + Q^2  ,
            
            where Q^2 = - q^2. The second equalities in (<ref>) and (<ref>) come from the identification of the states ω'^n and ψ^n with the states ρ^0,n. Similarly for the vector sector, we use the Feynman rules associated with the triple vertex (<ref>) and the vector meson propagator in  (<ref>) to extract the (elastic) EM form factors for the ρ meson, K^* meson and D^* meson :
            
            F_ρ^1 = F_ρ^2 = F_ρ(Q^2)    =   ∑_ng_ρ^n g_ρ^n ρρ/m_ρ^n^2 + Q^2  ,
            
            
            F_K^*^1 = F_K^*^2 = F_K^*(Q^2)    =    2 ∑_ng_ρ^n g_ρ^n K^* K^*/m_ρ^n^2 + Q^2  ,
            
            
            F_D^*^1 = F_D^*^2 = F_D^*(Q^2)    =    2 ∑_ng_ρ^n g_ρ^n D^* D^*/m_ρ^n^2 + Q^2  ,
            
            
            F_ρ^3 =  F_K^*^3  = F_D^*^3    =    0  .
            
            The electric, magnetic and quadrupole form
            factors are obtained using (<ref>).
            
            
            
             §.§ Low and high Q^2
            
            
            At low Q^2 the EM form factor of a pseudoscalar meson can be expanded as
            
                F_π^a(Q^2)= 1 -  1/6⟨ r_π^a^2⟩ Q^2 + ... ,
            
            where the second term defines the charge radius. A similar expression follows for the vector mesons.
            Notice that we have used the relation F_π^a(0)=1, which is due to charge conservation.
            In fact, the relation F_π^a(0)=1 follows nicely from the sum
            rules
            
            ∑_ng_ρ^n g_ρ^n ππ/m_ρ^n^2   =    2 ∑_ng_ρ^n g_ρ^n K K/m_ρ^n^2   =    2 ∑_ng_ρ^n g_ρ^n D D/m_ρ^n^2 = 1   .
            
            These sum rules can be proven using the equation and completeness relation of vector mesons as well
            as the normalization of the external states.  For the vector mesons the electric radius is obtained from the electric form factor as
            ⟨ r_V^a^2 ⟩ = -6 . d F_V^a^E(Q^2)/dQ^2| _Q^2=0  .
            
            The magnetic μ and quadrupole D moments of the vector
            mesons in our model take the canonical values
            
            μ   =    F_V^a^M(0) =  2   , 
            D      =   1/M_V^a^2F_V^a^Q(0)= - 1/M_V^a^2 ,
            
            where we have used the relation F_V^a(0)=1 which follows from the sum rules
            
            ∑_ng_ρ^n g_ρ^n ρρ/m_ρ^n^2   =    2 ∑_ng_ρ^n g_ρ^n K^* K^*/m_ρ^n^2   =    2 ∑_ng_ρ^n g_ρ^n D^* D^*/m_ρ^n^2=1  . 
            Again these sum rules follow from the equation and completeness of the ρ^n states and the normalization
            of the external states.  In fact, the sum rules Eqs. (<ref>) and (<ref>) are universal in
            bottom up and top-down holographic models for QCD. A discussion of these sum rules in the top-down approach
            can be found in Ref. <cit.>.
            
            In the regime of large Q^2, the EM form factors of pseudoscalar mesons can be expanded as
            
            F_π(Q^2)   =   1/Q^2∑_n=0^∞ g_ρ^ng_ρ^n ππ [ 1 - m_ρ^n^2/Q^2 + … ]
              , 
            F_K(Q^2)   =   2/Q^2∑_n=0^∞ g_ρ^ng_ρ^n K K  [ 1 - m_ρ^n^2/Q^2 + … ]  
            F_D(Q^2)   =   2/Q^2∑_n=0^∞ g_ρ^ng_ρ^n D D  [ 1 - m_ρ^n^2/Q^2 + … ] .
            A similar expansion holds for the EM form factors of vector mesons
            
                F_ρ(Q^2)   = 1/Q^2∑_n=0^∞ g_ρ^ng_ρ^n ρρ [ 1 - m_ρ^n^2/Q^2 + … ]
                  , 
                F_K^*(Q^2)   = 2/Q^2∑_n=0^∞ g_ρ^ng_ρ^n K^* K^* [ 1 - m_ρ^n^2/Q^2 + … ]
                  , 
                F_D^*(Q^2)   = 2/Q^2∑_n=0^∞ g_ρ^ng_ρ^n D^* D^* [ 1 - m_ρ^n^2/Q^2 + … ] .
            
            
            In the next section we present our predictions for the couplings and form factors involving the pions, kaons, ρ mesons, K^* mesons as well as the charmed mesons D and D^*. For the D and D^* EM form factors we  compare our results against data from Lattice QCD. Using
            (<ref>) and (<ref>) we will also extract the charge radii of all those mesons and compare against experimental data or lattice QCD data. Last but not least, the high-Q^2 behavior of the form factors in Eqs. (<ref>)-(<ref>) will be checked and compared with perturbative QCD calculations.
            
            
            
            
            
            § RESULTS AND COMPARISON WITH LATTICE QCD
            
            
            In this section we present our numerical results for the spectrum, decay constants, coupling constants and EM form factors  involving the charmed mesons. It is convenient to define unnormalized wave functions for the scalar mesons (ϕ̃_U^a,n and π̃_U^a,n), pseudoscalar mesons (ϕ_U^a,n and π_U^a,n), vector mesons (v_U^a,n) and axial vector mesons (a_U^a,n) so that the first coefficient in the near boundary expansion is fixed arbitrarily (due to the linearity of the differential equations). Eqs. (<ref>) and (<ref>) dictate the near boundary behavior of  the unnormalized wave functions:
            ϕ̃^a,n_U (z)    =     -z^2 + …    ,   π̃^a,n_U (z) =  -m_π_V^a,n^2/β_V^a(0)z^2 + … , ϕ^a,n_U (z)    =     -z^2 + …    ,   π^a,n_U (z) =  -m_π^a,n^2/β_A^a(0)z^2 + …  , 
            v_U^a,n(z)    =    z^2 + …    ,   
            a_U^a,n(z) = z^2 + …  . 
            In Eq. (<ref>), the first coefficients were fixed to 1 or -1 to guarantee a positive
            sign for the decay constants in Eqs. (<ref>)-(<ref>) and positive normalization constants.
            The normalized wave functions take the form
            
            ϕ̃^a,n(z)    =    N_π_V^a,nϕ̃_U^a,n(z)
                ,   π̃^a,n(z) = N_π_V^a,nπ̃_U^a,n(z)   , ϕ^a,n(z)    =    N_π^a,nϕ_U^a,n(z)
                ,   π^a,n(z) = N_π^a,nπ_U^a,n(z)   , v^n,a(z)   =    N_V^a,n v_U^a,n (z)    ,   
            a^n,a(z) = N_A^a,n a_U^a,n (z), 
            with the normalization constants defined by the integrals
            
            N_π_V^a,n^-2   =   ∫dz/z{ ( ∂_z  ϕ̃_U^a,n (z) )^2   +   β^a_V(z) (  π̃_U^a,n(z) -  ϕ̃_U^a,n(z) )^2}  , 
            N_π^a,n^-2   =   ∫dz/z{ ( ∂_z   ϕ_U^a,n (z) )^2   +   β^a_A(z) (   π_U^a,n(z) -   ϕ_U^a,n(z) )^2}  , 
            N_V^a,n^-2   =   ∫dz/z (v_U^a,n(z) )^2  , 
            N_A^a,n^-2   =   ∫dz/z (a_U^a,n(z) )^2  .
            
            The spectrum of vector mesons and scalar mesons is obtained by solving Eqs. (<ref>) and
            imposing the Neumann boundary conditions at the hard wall z=z_0. Similarly, the spectrum of axial vector
            mesons and pseudoscalar mesons is obtained by solving Eqs. (<ref>) and imposing Neumann
            boundary conditions at the hard wall.
            
            On the other hand, using Eqs. (<ref>) and (<ref>) we find that the meson decay
            constants, defined in Eqs. (<ref>)-(<ref>), are determined by the normalization constants through
            the relations
            
            
            f_π_V^a,n   =   2/g_5N_π_V^a,n   ,   
            f_π^a,n = 2/g_5N_π^a,n  , 
            g_V^a,n   =   2/g_5 N_V^a,n   ,   
            g_A^a,n = 2/g_5 N_A^a,n  .
            
            Having described the procedure for finding the meson spectrum and decay constants now we describe
            how we fit the parameters of our model, namely the quark masses m_u, m_s, m_c , quark condensates
            σ_u , σ_s, σ_c and the position of the hard wall z_0. We choose to fit the parameter
            z_0 using only the mass of the ρ meson, since that observable does not depend on any other parameter.
            We find z_0^-1 = 322.5  MeV. Then we proceed with a global fit for the quark masses and quark
            condensates using 10 observables, namely the light meson masses (m_π,m_a_1), the strange meson
            masses (m_K,m_K^∗,m_K_1,m_K_0^*) and the charmed meson masses
            (m_D,m_D^∗,m_D_s,m_D^*_s). Note that we have included the scalar meson K_0^*,
            which is associated with flavor symmetry breaking. Numerically, we find the best global fit for the
            parameters  m_u=9 MeV, m_s=190 MeV and m_c=1560 MeV for the quark masses;
            σ_u=(198 MeV )^3, σ_s=(205 MeV)^3 and σ_c=(280 MeV)^3
            for the quark condensates. In Table <ref> we compare the model fit to the observables with
            their experimental values.
            
            
            
            
            Note that the fit works very well for the isospin and strange sectors, as already known from
            previous works. The extension of the model to the charm sector also gives a reasonably good fit
            of properties of heavy-light mesons, like the D and D^∗ mesons.
            
            
            
            Once we have fitted the parameters of the model, we are able to make predictions.
            In Table (<ref>) we show a set of predictions for masses and decay constants.
            In the cases where experimental or lattice data is available the measured values are
            presented. Regarding the masses m_D^∗_0 and m_D^∗_0s, the large difference between
            the model prediction and the experimental values <cit.> may be related to the not
            clear distinction between the ground and excited states.
            
            
            
            Now we move to the triple meson couplings defined in the effective Lagrangians (<ref>) and (<ref>).
            Using the dictionary  (<ref>, <ref>) and the relations (<ref>) we find the couplings g_ρ^n ππ, g_ρ^n ρρ, g_ρ^n K K, g_ρ^n K^* K^*, g_ρ^n D D and
            g_ρ^n D^* D^*. The results are shown in Table <ref> where we notice an interesting
            feature taking place. The triple coupling g_ρ^n D D, involving the heavy-light pseudoscalar charmed
            mesons, does not decrease with n in the same way as the  triple couplings  g_ρ^n ππ and
            g_ρ^n K K, involving light and strange pseudoscalar mesons respectively. The same behavior appears
            in the triple coupling g_ρ^n D^∗ D^∗, involving vectorial charmed mesons,  when compared to
            the triple couplings g_ρ^n ρρ and g_ρ^n K^* K^*, involving vectorial light and strange
            mesons respectively. In the case of g_ρ^n D D we see that actually the first resonance ρ^n=1
            couples stronger than the fundamental ρ^n=0.
            
            
            
            Due to flavor symmetry breaking, through the different values for the quark masses and condensates, we expect a violation of  SU(4) relations given in (<ref>)-(<ref>). We compare in Table <ref> our
            results with the expectations from the SU(4) flavor symmetric case. We note that we find the trend
            g_ρ D D < g_ρ K K < g_ρππ/2 for the pseudoscalar mesons, which opposite to that found with the QCD sum rules <cit.>
            and Dyson-Schwinger calculations in Ref. <cit.>, but it agrees with calculations based
            ^3P_0 quark-pair creation model in the nonrelativistic quark model of Refs. <cit.>. For the vectorial mesons we find a similar trend g_ρ D^* D^* < g_ρ K^* K^* < g_ρρρ/2. Note, however, that the coupling g_ρ K^* K^* is very close to g_ρρρ/2. The reason behind this proximity is that the vector meson spectrum, found from the first equation in (<ref>), depends more on the condensate difference than the mass difference appearing in β_V^a(z). Since the strange and light condensates σ_s and σ_u are very close to each other the masses and wave functions of the ρ and K^* are very similar.
            
            
            
            
            
            
            Finally, we show our results for the meson (elastic) EM form factors defined in the previous section.
            Using the couplings obtained in Table <ref> and evaluating the expressions
            in Eqs. (<ref>)-(<ref>) we obtain a series expansion for the EM form factors
            of the pion, kaon and D meson. In a similar fashion, we take the coupling in Table <ref> and
            evaluate the expressions in Eqs. (<ref>)-(<ref>) to obtain a series expansion
            for the EM form factors of the ρ, K^* and D^* mesons. In both cases, a good convergence is achieved
            after considering 5 states (4 resonances ρ^n besides the fundamental ρ^n=0). This is explicitly shown
            in Table <ref> for the case Q^2=0.
            
            
            
            Table <ref> reveals again a clear distinction between the light mesons and charmed mesons.
            In the former, vector meson dominance (VMD) is a good approximation, whereas in the case of the latter
            the EM form factors receive substantial contribution from the first ρ resonance. This is a nice
            example of generalized vector meson dominance (GVMD) in EM form factors. This is consistent with
            Ref. <cit.>,  where the authors claimed that the radial excitations of the ρ meson
            are important for EM  form factors of nonzero spin hadrons, as nucleons.
            
            
            
            In Figs. <ref>, <ref>, <ref> and <ref> we show our results for the EM form factors of the π, ρ, K and K^* mesons. The pion and kaon EM form factors are compared against experimental data. Previous results for the pion and kaon EM form factors in holographic QCD can be found in <cit.> and <cit.> respectively. Previous results for the ρ meson EM form factor can be found in <cit.>. Finally we show in Figs. <ref> and <ref> our results for the D meson and D^∗ meson EM form factors compared with data from lattice QCD. As promised, we find a reasonable agreement between our model and the lattice results.
            
            
            
            At low Q^2, we use the relations in Eqs. (<ref>) and (<ref>) to extract
            the charge radii of pseudoscalar and vector mesons. In Table <ref> we compare our results
            for the light mesons against experimental data and those for the charmed mesons against lattice QCD data.
            
            
            
            We use the expansions in Eqs. (<ref>) and (<ref>) to obtain the Q^2
            dependence of the EM form factors; the results are shown in Figs. <ref> and
            <ref>. For the case of  pseudoscalar mesons, we find the behavior F_π^a(Q^2) ∼ Q^-2. For the vector meson EM form factors we find that F_V^a(Q^2) ∼ Q^-4. Both results are
            consistent with perturbative QCD expectations and conformal symmetry in the UV.
            
            
            
            
            § CONCLUSIONS
            
            
            We have extended the two-flavor hard-wall holographic model of Ref. <cit.> to four
            flavors. By fitting the seven parameters of the model, which are three quark masses, three condensates
            and the hard-wall scale z_0, to eleven selected meson masses, the model provides a good description
            of weak decay constants of more than a dozen of light and strange and charmed mesons. We have also
            investigated the effects of flavor symmetry breaking on three-meson couplings and form factors.
            In particular, we have made predictions for the strong couplings g_ρ^nππ, g_ρ^nρρ,  g_ρ^n K K, g_ρ^n K^* K^*,  g_ρ^n D D and g_ρ^n D^*D^*. Moreover, using our results for those couplings we have been able to
            evaluate the π, ρ, K, K^*, D and D^* electromagnetic form factors. For the D and D^*  electromagnetic form factors we found a reasonable agreement with the lattice QCD results of Ref. <cit.>.
            
            Our results for the  couplings involving the ground-state ρ meson and the charmed mesons,
            namely g_ρ DD and g_ρ D^* D^* are smaller than the SU(4) symmetry values, as shown
            in Table <ref>. Our result g_ρ DD=1.103 is also smaller than
            predictions based on the VMD model <cit.> where g_ρ D D = 2.52 - 2.8. Moreover,
            we found that g_ρ DD < g_ρππ/2, which is of the opposite trend to the predictions
            based on QCD sum rules <cit.> and Dyson-Schwinger equations of QCD <cit.>,
            but it agrees with that obtained with the  ^3 P_0 pair-creation model in the nonrelativistic quark model
            of Refs.<cit.>. A possible explanation for the discrepancy for the small values
            of the couplings is that the electromagnetic form factor of the D meson is a dramatic example where the
            VMD approximation is broken and the contribution from the resonances ρ^n can not be neglected. It is
            interesting to notice the relation between the breaking of the VMD approximation and the breaking of the
            SU(4) symmetry. In a VMD approximation we would find that
            
            
            2 g_ρ D D = m_ρ^2/g_ρ = g_ρππ .
            
            The first equality comes from applying VMD to the D isospin form factor; in our framework, this relation
            also comes from the EM form factor. The second equality is the well known VMD result for the pion EM
            form factor.  The relation in Eq. (<ref>) can be extended to other couplings and it means
            that a VMD approximation necessarily implies a universality between the couplings. Interestingly, the
            result in Eq. (<ref>) for the coupling g_ρ DD matches with the SU(4) symmetry
            expectations. Then it is reasonable to interpret a dramatic breaking of the SU(4) flavor symmetry
            in terms of a dramatic breaking of the VMD approximation, which is exactly what we have found for the
            charmed mesons D and D^*.
            
            We finish this paper by reiterating our earlier remarks on the applicability of our model. Our holographic
            QCD model is based on an extensions of a light-flavor chiral Lagrangian, which should be adequate
            to describe heavy-light mesons, as the internal structure of these mesons is governed by essentially the
            same nonperturbative physics governing the internal structure of light mesons, which occurs at the scale
            Λ_ QCD.   On the other hand, the internal structure of heavy-heavy mesons, as the
            ψ and η_c mesons, is governed by short-distance physics at the scale of the heavy quark
            mass. An appropriate holographic description of such mesons most likely requires the inclusion of
            long open strings. In that scenario, it should be possible, in particular, to describe the non-relativistic
            limit of heavy quarks where a spin-flavor symmetry emerges <cit.>. Although there have
            been some interesting top-down <cit.> and
            bottom-up <cit.> proposals, a realistic model for heavy-heavy mesons
            remains a challenge in holographic QCD.
            
            In holographic QCD, it is assumed that the quark mass coefficient m_q in the near boundary expansion of the classical field X_0(z) behaves as the source of the operator q̅(x)q(x). Then the holographic dictionary leads to conclude that the parameter σ is also in one to one correspondence with the vacuum expectation value (v.e.v.) ⟨q̅(x)q(x) ⟩. This matching, however, is ambiguous, as discussed in Ref. <cit.>, because ⟨q̅(x)q(x) ⟩ is actually a scale-dependent quantity whereas m_q and σ are obtained from a global fit to the meson spectrum. This issue actually becomes exacerbated as the quark mass increases.
            
            In QCD, the quantity ⟨q̅(x)q(x) ⟩ is identified with the trace of the quark propagator S, i.e. ⟨q̅(x)q(x) ⟩ = -  Tr  S(x-x) 
            = -  Tr  S(0). It contains a nonperturbative, low-energy contribution from a dynamical component of chiral symmetry breaking, and an essentially perturbative contribution due to the explicit chiral symmetry breaking driven by the quark mass. In the heavy quark mass limit, the perturbative contribution dominates and the nonperturbative contribution goes to zero. For that reason, and to make contact 
            with the traditional definition of the quark condensate in  QCD sum rules <cit.>, in lattice simulations the perturbative contribution is subtracted; see e.g. <cit.>. Interestingly, 
            the authors of Ref. <cit.> found, after subtracting the 
            perturbative contribution, that the strange quark condensate at the MS scale of 2  GeV  is larger than that of the light quarks. So far there are no such lattice calculations for the charm and bottom quark, 
            but calculations within the framework of Dyson-Schwinger equations <cit.> find that the nonperturbative component of chiral symmetry breaking decreases 
            with increasing current-quark mass, as expected. 
            
            Our results, obtained from a global fit to the meson spectrum, indicate that σ 
            increases with m_q. Although, as discussed above, the relation between σ and the 
            QCD v.e.v. ⟨q̅(x)q(x) ⟩ is far from clear, one could assume that relation
            as being strictly one-to-one and conclude that ⟨q̅(x)q(x) ⟩ increases with the quark mass unless a perturbative subtraction is also implemented in the holographic model. For the case of the charm quark this means that a large value for σ_c does not necessarily imply a large charm quark condensate. There is an additional issue that requires further study. In QCD, the v.e.v. of the operator q̅ q, with canonical dimension 
            Δ=3, is expected to acquire a large anomalous dimension in the infrared. In our holographic 
            model, we have made the ad hoc approximation of keeping the same canonical dimension for ⟨q̅ q ⟩. If we take into account anomalous dimension effects, 
            corrections to m_q and σ are expected. We hope to pursue this line of research in the near future.
            
            It is also important to bear in mind that results from a holographic QCD approach are supposedly
            referring to leading-order in an expansion of 1/N_c and in the large 't Hooft coupling λ=g_YM^2 N_c.
            As such, loop corrections for the hadronic propagators and vertices are not taken into account.
            The 1/N_c and/or 1/λ corrections to the effective chiral-flavor Lagrangians
            in holographic QCD is a fascinating open problem and deserves further studies.
            
            
            
            
            § ACKNOWLEDGEMENTS
            
            
            Work partially supported by Fundação de Amparo à Pesquisa do Estado de
            São Paulo (FAPESP), Grants No. 2015/17609-3 (A.B.-B.) and 2013/01907-0 (G.K.),  
            Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Grant
            No. 305894/2009-9 (G.K.) and Coordenação de Aperfeiçõamento de Pessoal de Nível
            Superior (CAPES) for a doctoral fellowship (C.M).  A.B-B also 
            acknowledges partial financial support from the  grant 
            CERN/FIS-NUC/0045/2015. The authors also thank K.U. Can for providing them
            the lattice data for the electromagnetic form factors of Ref. <cit.>.
            
            
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