Semileptonic D meson decays to the vector, axial vector and scalar mesons in Hard-Wall AdS/QCD correspondence

In this work, a Hard-Wall AdS/QCD model with 4 flavours is utilized to calculate the transition form factors for the semileptonic D→(V,A,S)ℓνℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D\rightarrow (V, A, S)\,\ell \,\nu _{\ell }$$\end{document}. These decays occur by c→dℓνℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c \rightarrow d\,\ell \,\nu _{\ell }$$\end{document} transition at quark level for D0→ρ-(b1-,a1-,a0-)ℓνℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{0} \rightarrow \rho ^{-}(b_{1}^{-}, a_{1}^{-}, a_{0}^{-})\,\ell \,\nu _{\ell }$$\end{document}, Ds+→K0(K1A0,K1B0)ℓνℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_{s}^{+} \rightarrow K_{0}(K_{1A}^{0}, K_{1B}^{0})\,\ell \,\nu _{\ell }$$\end{document} decays while the semileptonic decays D0→K∗-(K1A-,K1B-,K0-)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{0} \rightarrow K^{_{*-}}(K_{1A}^{_-}, K_{1B}^{_-}, K_{0}^{_-})$$\end{document}ℓνℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell \,\nu _{\ell }$$\end{document} and D+→K¯0(K¯1A0,K¯1B0)ℓνℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{+} \rightarrow {\bar{K}}_{0}({\bar{K}}_{1A}^{0}, {\bar{K}}_{1B}^{0})\,\ell \,\nu _{\ell }$$\end{document} proceed by c→sℓνℓ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c \rightarrow \, s\, \ell \,\nu _{\ell }$$\end{document} transition. The masses and decay constants of scalar mesons as well as the form factors and the branching ratios of aforementioned decays are calculated in our model, and a comparison is also made between our results and predictions of other theoretical methods and the existing experimental values.


Introduction
Charm mesons are the lightest particles involve a c quark therefore, their decays are good tools for the study of the weak interactions. In recent years, experimental development has been achieved in considering semileptonic decays of these groups of mesons. The most valuable measurements are reported from BES III. In this collaboration analysis are presented for D + →K 0 e + ν e , D + → π 0 e + ν e , D 0(+) → π −(0) μ + ν μ , D + s → φ e + (μ + ) ν e(μ) , D + s → η(η ) μ + ν μ , D + → η(η ) e + ν e , D 0 → K − μ + ν μ , D + s → K 0( * 0) e + ν e , D 0 →K 0 π − e + ν e , D + → f 0 (500) e + ν e and D → ρ e + ν e decays [1][2][3][4][5][6][7][8][9]. In addition, the data extracted from Belle II collaboration are used to evaluate important observable for D * + s → D + s γ , D + s → μ + (τ + ) ν μ(τ ) , D * + → D 0 π + , D 0 → K − (π − ) + ν and D 0 → ν ν decays [10][11][12]. On the other hand, Theoretical Studies of D mesons semileptonic decays are helpful tools to: a e-mail: samira.momeni@ph.iut.ac.ir (corresponding author) b e-mail: saghebfar@mut-es.ac.ir The first step in studying hadronic decays is to evaluate form factors, which can give an overview from distribution of hadronic matter and the strong interaction between particles involved in the transition. The form factors are functions of q 2 where q is the transfer momentum. Having form factors, theoretical predictions for many observable such as branching fraction, CP asymmetry, Isospin asymmetry and Forward-Backward asymmetry can be done. D meson transitions are studied via different approaches. The covariant confined quark model (CCQM) framework is utilized to evaluate form factors of D + → (D 0 , ρ 0 , ω, η, η ) + ν and D + s → (D 0 , φ, K 0 , K * 0 , η, η ) + ν decays in [13,14]. The form factors of D → π(K , ρ) ν decays are estimated via Light-Cone QCD Sum Rules (LCSR) approach in [15][16][17][18]. In the LCSR method, operator product expansions (OPE) on the light-cone are combined with QCD sum rule techniques in the region where q 2 is near zero. In [19], the experimental measurements are used to determine the form factors of D → K ν transition. The semileptonic processes D → π, ρ, K and K * have been investigated by the heavy quark effective theory (HQET) in Ref. [20], and the form factors of the D → π(K , K * ) ν decays have been calculated by the lattice QCD (LQCD) approach in Refs. [21][22][23]. The LQCD can be used to determine the form factors in The main purpose of this paper is the form factor investigation for the semileptonic D → (V, A, S) ν decays with D = (D 0 , and = e, μ, in a Hard-Wall AdS/QCD model with N f = 4. In the model that is used in this study, we can predict the form factors of charm meson decays in all of the physical range of q 2 , which is an advantage of the AdS/QCD model in studying charm meson decays. This paper is organized as follows: the presented model, including scalar, pseudoscalar, vector and axial vector mesons, is introduced in Sect. 2. The wave functions and the decay constant of aforementioned mesons are also extracted from the model in this section. The form factors for the semileptonic decays of D → (V, A, S) ν are derived in Sect. 3., and the numerical analysis for the masses and wave functions of mesons as well as the form factors and branching ratios for considered semileptonic decays are described in Sect. 4. For a better analysis, a comparison is made between our estimations and the results of other methods and existing experimental values. Finally, Sect. 5 is reserved for our conclusion and discussions.
2 The Hard-Wall AdS/QCD model involving scalar, pseudoscalar, vector and axial vector mesons: wave functions, decay constant and masses The first step, in our framework, to evaluate form factors or strong couplings is to estimate wave functions in 5 dimensions for the mesons included in the model. For this aim, the Anti-de Sitter space metric is chosen as: where M, N = 0, 1, 2, 3, 4 and η M N = diag(1, −1, −1, −1, −1). Moreover, the warp factor is shown with R, which can be chosen as R = 1 in pure AdS [69]. In Hard-Wall AdS/QCD, the AdS space is compactified by two different boundary conditions on radial coordinate z. The UV boundary at z = ε → 0 corresponds to the UV limit of QCD, and the conformal symmetry of QCD is broken by locating a wall at z = z m , which also simulates the confinement feature of QCD. According to the general philosophy of the AdS/QCD, every operator in the 4D field theory corresponds to a 5D source field in bulk. Here, the correspondences are: where L(R) μ,a , X and q are the N f left-handed (righthanded) gauge, scalar and quark field, respectively. Moreover, q L/R = (1 ± γ 5 ) q and for SU(N f ) group t a (with a = 1, . . . N 2 f − 1) are the generators by the trace condition Tr(t a t b ) = 1/2 δ ab . In this paper, the 5D action with SU(N f ) L ⊗ SU(N f ) R symmetry is considered as [69]: where the field strengths are defined by: with L M = L a M t a and R M = R a M t a . Moreover, the nonabelian gauge fields and the scalar one interact through the covariant derivative D M X = ∂ M X −i L M X +i X R M . Vector (V) and axial vector (A) fields can be written in terms of righthand gauge fields as V = (L + R)/2 and A = (L − R)/2. In the above definitions, N f is the number of flavours, and here we take N f = 4, which means all of the light, strange and charmed mesons are included in our model. The scalar field X can be expanded in exponential form as: where the background part is denoted by X 0 and π are (pseudoscalar) fluctuations. For the classical solution, we turn off all fields except X 0 and solve the equation of motion. Finally, we arrive at: where M denotes the quark-mass matrix and is the quark condensates qq one. Moreover, ζ = √ N c /2π is the normalization parameter [70]. With flavour symmetry, Eq.(5) can be written as X = e 2iπ a t a X 0 , which is used to predict masses and decay constants of ground-state and excited light and strange mesons in [71,72].
To evaluate the wave functions of the particles included in this study, the action Eq. (3) must be expanded up to second order in vector (V), axial vector (A) and pseudoscalar field (π ) as: in [68], are collected in the Appendix. In the following subsection, we consider the axial sector and the vector one from action (7) to study the wave functions, decay constant and masses of physical particles.

Axial sector
The axial vector field ( A N ) satisfies the equation of motion as: where we have defined: Now in Eq. (9), we write A a N = (A a μ , A a z ) and make the decomposition A a μ = A a μ⊥ + A a μ . For the transverse part, which describes the axial vector states, ( A a μ⊥ ), we can write: in a gauge where A a z = 0. Here q is the Fourier variable conjugate to the 4-dimensional coordinates, x. We shall write the transverse part of the axial vector field in terms of its boundary values at UV (A 0a μ⊥ ) multiplying bulk-to-boundary propagator (A a ), A a μ⊥ (q, z) = A 0a μ⊥ (q)A a (q 2 , z). A a (q 2 , z) satisfies the same equation as A a μ⊥ (q, z) with the boundary conditions A a (q 2 , ε) = 1 and ∂ z A a (q 2 , z 0 ) = 0. The longitudinal part of the axial vector field, defined as A a μ = ∂ μ φ a and π a , describe the pseudoscalar fields and satisfy the following equations: with the boundary conditions, φ a (q 2 , ε) = 0, π a (q 2 , ε) = −1, and ∂ z φ a (q 2 , z 0 ) = ∂ z π a (q 2 , z 0 ) = 0. In general, the form of differential equations Eqs. (11,12,13), A a (q 2 , z), φ a (q 2 , z) and π a (q 2 , z) can be solved numerically. Using Green's function formalism to solve Eqs. (11,12,13), the bulk-to-boundary propagator A a , π a and φ a can be written as a sum over axial vector and pseudoscalar mesons poles as: P n f a P n φ a n (z) q 2 − m a2 P n , π a (q 2 , z) = n −g 5 m a2 P n f a P n π a n (z) where f A n and ( f P n ) are decay constants of the n th axial vector meson and the pseudoscalar one and for the n th axial vector meson's wave function are ψ A n ( ) = 0 and ∂ z ψ A n (z 0 ) = 0 are the boundary conditions. Moreover, for the pseudoscalar meson's wave functions, the boundary conditions are: φ a n (ε) = π a n (ε) = 0 and ∂ z φ a n (z 0 ) = ∂ z π a n (z 0 ) = 0 and the normalization condition for these wavefunctions is (dz/z)g(z) 2 = 1 for g = ψ a A n , π a n , φ a n . The decay constants of the n th mode of the axial vector and the pseudoscalar states are related to their nth wave functions as [69]: With N f = 4, the axial vector A and the pseudoscalar π involves the light, strange and charmed states can be written as follows: The physical states K 1 (1270) and K 1 (1400) mesons are related to K 1A and K 1B states in terms of a mixing angle θ K as the following terms: Various approaches have been utilized to estimate the mixing angle θ K by the experimental data. The result 35 • < |θ K | < 55 • was found in Ref. [73], and two possible solutions with |θ K | ≈ 33 • and 57 • were reported in Ref. [74]. Moreover, the value θ K = −(34 ± 13) • is obtained via analyzing B → K 1 (12070)γ and τ → K 1 (12070) ν τ data in [75].

Vector sector
The equation of motion for the vector field (V N ) is similar to Eq. (9) by replacing the V N → A N and β a → α a (z) and applying ∂ μ V a μ⊥ (x, z) = 0, for the transverse part of the vector field, which represents the vector meson states, the following result is obtained: where V a (q 2 , z) satisfies the same equation as (19) with the boundary conditions V a (q 2 , ε) = 1 and ∂ z V a (q 2 , z 0 ) = 0. The longitudinal parts of the vector field, defined as V a μ = ∂ μ ξ a and V a z = −∂ zπ a , describe the scalar states and are coupled as follows: where ξ a =φ a −π a . The boundary conditions for φ a and π a areφ a (q, ε) = 0,π a (q, ε) = −1 and ∂ zφ For V a ,φ a andπ a , the Green functions are: The decay constants of the n th mode of the vector meson and the scalar one have been obtained as [69]: Finally, the SU(4) vector V and pseudoscalarπ meson matrices in terms of the charged states are introduced as follows:

D → (V, A, S) ν form factors
In this section, the form factors of D → (V, A, S) ν are derived in Hard-Wall AdS/QCD model. At the quark level, this process is induced by the semileptonic decay 1A ,K 0 1B ) ν processes are considered. In the standard model of particle physics, the semileptonic decays D → M ν with M = (V, A, S) are described by the following effective Hamiltonian: where we take α(V, S) = 1 and for M = A, α(M) = −1.
Moreover, G F is the Fermi constant, and V cq is the CKM matrix elements. The decay amplitude for these decays is obtained by inserting Eq. (24) between the initial meson D and final state M as: where p 1 and p 2 are the momentum of D and M states, respectively. To evaluate the decay amplitude (M), we need to calculate the matrix elements M( p 2 )|q γ μ c|D( p) and M( p 2 )|q γ μ γ 5 c|D( p) . These matrix elements can be parameterized in terms of the form factors. For M = V , these matrix elements are defined as: where ε V is the polarization vector of V meson and q = p 1 − p 2 is the transport momentum. In Eqs. (26) and (27), can be written as a linear combination of A 1 (q 2 ) and A 2 (q 2 ) as: 3) are defined as: where ε A is the polarization vector of the axial-vector meson And finally, the form factors of D → S ν are defined as: To evaluate the form factors of D → (V, A, S) ν in Hard-Wall AdS/QCD, we start with the correlation function, including the currents of 3 involving particles. These 3-point functions can be obtained by functionally differentiating the 5-D action with respect to their sources, which are taken to be boundary values of the 5-D fields that have the correct quantum numbers as [36,37,76,77]: where f V , f P , f A and f S are the decay constants of the vector, pseudoscalar, axial vector and scalar mesons, respectively. In this step, the following results can be obtained: wherê are defined in our notations. Moreover, f O i is the decay constant of the i th meson, and the limit ( is taken in the final result. Now, we need the relevant actions presented in Eqs. (42,43,44,45,46 ). For example, two vector mesons and one pseudoscalar state are separated from the total action for S(VVP). One vector meson, one axial vector and one pseudoscalar meson are also considered for S(VAP). These relevant actions are calculated in our recent paper as [68]: where f abc is the SU (4) structure constant and the terms containing this factor are generated from the gauge part of the main action. The following definitions are also used in Eqs. (48)(49)(50)(51): The values of f abc are taken from [78]. Moreover, the other values of k abc , h abc and g abc , which are used in the numerical part, are collected in the Appendix. Using the Fourier transforms as [58,79]: the form factors of D → V ν can be obtained as: and for D → A ν we have: Finally, the form factors of D → S ν are resulted as:

Numerical analysis
The numeric analysis for the masses, decay constants, and the wave functions of scalar mesons are presented in this section. Moreover, form factors and the branching ratios of D meson decays into vector, axial vector and scalar mesons are evaluated.  3 . The wave functions, masses, and decay constants of some vector, axial vector, and the pseudoscalar mesons are studied in [68], and here we focus on the scalar mesons. The results for the masses and the decay constants of the ground state scalar mesons a − 0 , K − 0 , K 0 , D 0 and D − 0s are listed in Table  1. In this table, the experimental values of masses as well as the QCD Sum Rule (QCDSR) predictions for the decay constants are also presented. The masses are taken from [80] and the decay constants are given in [81][82][83]. It should be noted that for neutral scalars a 0 0 , σ and χ c0 , the differential equations take the form ∂ zφ a (q 2 , z) = 0, and considering Eq. (23), for these states, the decay constant becomes zero. This result was predictable according to charge conjugation invariance or conservation of vector current.

Wave functions and decay constants
It is possible to estimate the wave functions with the masses of physical states. The wave functionsφ S andπ S for the ground states S = (a − 0 , K − 0 , K 0 , D 0 , D − 0s ) are shown in Fig. 1 as functions of z/z m . The dash, dotted, dash-dotted, dash-dot-dot and short-dash lines are utilized forφ 1 andπ 1 of a − 0 , K − 0 , K 0 , D 0 and D − 0s , respectively.

Form factors
To estimate the obtained form factors in Sect. 3, we need to know the masses of the initial states (D) and those of vector (V ), axial vector (A), and scalar meson (S) as the final states. The used masses in numerical analysis are given in Table 2.
The values for masses of ρ − , K * − , D + and a − 1 are taken from [80], while our model in [68] predicts the masses of D 0 , D + s , K − 1A . Moreover, for b − 1 and K − 1B the predictions of the sum rule approach are inspired from [84].
At this point, we can estimate the form factors for each aforementioned semileptonic decay. The obtained results of the estimation for the form factors , for D → A ν decays, and f + , for D → S ν transitions, at q 2 = 0, are presented in Table 3. Note that for D → S ν decays, we consider f + (0) = f − (0). In the predictions for the form factors, the main uncertainty comes from the masses of the initial and final mesons, and the quark condensates σ q .
The form factors of D → (V, A, S) ν decays are calculated with different theoretical frameworks. To compare the different results, we rescale them according to the form factor definitions in Eqs. (27,30,32). Table 4 shows the values of the rescaled form factors at q 2 = 0 for D → V ν decays, according to different theoretical approaches such as the LCSR [17], the 3PSR [27], the heavy quark effective field theory (HQETF) [85,86], the relativistic harmonic oscillator potential model (RHOPM) [87], the quark model [88,89], the light-front quark model (LFQM) [90], the heavy meson and chiral symmetries (HMχ T) [91] and the LQCD [92]. The experimental measurements for the form factors of D 0 → ρ − ν , reported in CLEO [93], are also included in Table 4. In Table 5, a comparison is made between the estimation for the form factors of D → A ν decays obtained from this work and the 3PSR [34], the LCSR [18] and the LFQM [94] predictions. It is essential to note that for the V (q 2 ) and A(q 2 ) form factors, the presented parameterizations in Eqs. (26) and (29) are different from other methods, and we can not compare these form factors. Table 1 Predictions for the masses and the decay constants of a − 0 , K − 0 , K 0 , D 0 and D − 0s scalar mesons. The experimental values for messes are reported in [80], while the QCDSR predictions for the decay constants are taken from [81][82][83] Observable  For the D → S ν category, only theoretical methods are used to study D 0 → a − 0 ν . As can be noticed from Table 3,  To obtain the form factors of D → K 1 (1270) ν using Eqs. (17,29,30 ) the following relations are obtained:  0.57 ± 0.07 Table 4 Transition form factors A 1 , A 2 and A 0 of the D → V ν at q 2 = 0 in our model and other theoretical approaches where on the right hand side of Eqs. (69-72), K 1 = K 1 (1270) and r M = m D /m M . To evaluate the form factors of D → K 1 (1400) ν , the replacements of sin θ K → cos θ K , cos θ K → − sin θ K , and K 1 → K 1 (1400) are required.

Branching ratios
To evaluate the branching ratio values for the D → (V, A, S) ν decays, the decay amplitude in Eq. (25), and definitions for the form factors given in Eqs. (26,27,29,30,32) are required. So, the differential decay widths are found Table 5 The results for the transition form factors V 1 , V 2 and V 0 of D → A ν decays at q 2 = 0, as well as predictions of 3PSR [33,34], LFQM [94] and LCSR [18] Table 7 Branching ratio values of the semileptonic D → A ν decays in Hard-Wall AdS/QCD, LCSR [18] and 3PSR [33,34]. For D → K 1 ν with K 1 = (K 1 (1270), K 1 (1400)), the values of branching ratios are reported at θ K = −(34 ± 13) • In summary, the form factors of the semileptonic charm mesons decay into the light and strange vector, axial vector and scalar mesons are evaluated in Hard-Wall AdS/QCD including 4 flavours (u, d, s, c). The wave functions of the scalar mesons, the masses and the decay constants of this group are studied in detail, and our predictions for the masses  Table 10 The values of g abc , h abc , l abc and k abc which are used in numerical analysis (a, b, c) g abc h abc l abc k abc (2,9,11) are compared with the experimental data. Moreover, the decay constants are compared with the results of the sum rule method. Our estimations for the form factors at q 2 = 0 are compared with other theoretical frameworks such as: LCSR, 3PSR, HQETF, RHOPM, QM, LFQM, HMχ T and LQCD, as well as the measurements of CLEO collaboration.
According to the presented definitions for the matrix elements in terms of form factors, the formulas for the decay widths are obtained. The branching ratios are estimated for the mentioned model and the results are compared with the experimental data and the predictions of the other theoretical approaches. The prediction for the branching ratio of the D 0 → ρ − ν decay is in good agreement with the reported results of LCSR, HQEFT, LFQM, and the experimental values. For the semileptonic D 0 → K * − ν decay, the estimation for the branching ratio value has a better agreement with the experimental report. The predicted form factors in this study can be used to evaluate the branching fraction of nonleptonic and forward-backward asymmetry of leptonic D meson decays. This model can also be extended to 5 flavours with q = (u, d, s, c, b) to study the B meson decays.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: In Table 6, the results of experimental data from CLEO(2013) [93] and CLEO(2005) [98] are considered and compared with the obtained theoretical results in this work.] Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copy-  The nonzero values for g abc , h abc , l abc and k abc used in numerical analysis are given in Table 10.