Twist-2 Pseudoscalar and Vector Meson Distribution Amplitudes in Light-Front Quark Model with Exponential-type Confining Potential

We study the twist-2 distribution amplitudes (DAs) and the decay constants of pseudoscalar light ($\pi$, $K$) and heavy ($D$, $D_s$, $B$, $B_s$) mesons as well as the longitudinally and transversely polarized vector light ($\rho$, $K^*$) and heavy ($D^*$, $D_s^*$, $B^*$, $B_s^*$) mesons in the light-front quark model with the Coulomb plus exponential-type confining potential $V_{\rm {exp}} = a + b e^{\alpha r}$ in addition to the hyperfine interaction. We first compute the mass spectra of ground state pseudoscalar and vector light and heavy mesons and fix the model parameters necessary for the analysis, applying the variational principle with the trial wave function up to the first three lowest order harmonic oscillator (HO) wave functions $\Phi(x, \textbf{k}_\bot) = \sum_{n=1}^{3} c_n \phi_{nS}$. We then obtain the numerical results for the corresponding decay constants of light and heavy mesons. We estimate the DAs, analyze their variation as a function of momentum fraction and compute the first six $\xi$-moments of the $B$ and $D$ mesons as well. Our results are well in agreement with the available experimental data both for the light and heavy mesons. We also compare them with other theoretical model predictions.


I. INTRODUCTION
In the last few decades, the light as well as heavy quark mesonic systems have provided a great deal of important and attractive information on the precise determination of the fundamental parameters of the Standard Model (SM). The nonperturbative structure of the hadron is well described by the hadronic or quark distribution amplitudes (DAs) which not only encode important information on bound states in QCD but also play an essential role in describing the various hard exclusive processes [1,2] of QCD via the factorization theorem [3] in analogous to parton distributions in inclusive processes.
One of the most successful and efficient nonperturbative approaches is the LFQM which has been widely used in the phenomenological studies of the hadron physics. It takes advantage of the equal light-front time (τ = t + z/c) quantization and includes the important relativistic effects in the hadronic wave functions [97][98][99] that are neglected in the traditional equal-time Hamiltonian formalism. Apart from having the maximum number (seven) of interaction free (or kinetic) generators, the rational energy-momentum dispersion relation p − = (p 2 ⊥ + m 2 )/p + yields the sign correlation between the light-front energy p − (= p 0 − p 3 ) and the light-front longitudinal momentum p + (= p 0 + p 3 ) leading to the suppression of vacuum fluctuations with the decoupling of complicated non-trivial zero modes. Therefore, a clean Fock state expansion of hadronic wave functions based on a simple vacuum can be built [100][101][102][103]. The light-front wave function (LFWF) can be expressed in terms of hadron momentum-independent internal momentum fraction variables making it explicitly Lorentz invariant [104]. Based on these properties, the LFQM has been developed and successfully applied to evaluate various meson phenomenologies such as the mass spectra of heavy and light mesons, their decay constants, DAs, form factors, generalized parton distributions (GPDs). While the direct connection between the LFQM and QCD has still not been established, recent results on the analyses of twist-2 and twist-3 quark-antiquark DAs for pseudoscalar and vector mesons in LFQM [105] indicated that the constituent quark and antiquark could be considered as dressed constituents including the zero-mode quantum fluctuations from the QCD vacuum. Also, the light-front holography based on the five-dimensional anti-de Sitter (AdS) space-time and the conformal symmetry has helped in understanding the nature of the effective confinement potential and the resulting light-front wave functions for both light and heavy mesons [106]. The results on the LFQM analysis of the pion form factor in both spacelike and time-like regions [107] are found to be compatible with the holographic approach to light-front QCD [108]. These developments motivate us to thoroughly analyse the ground state pseudoscalar and vector mesons mass spectra, decay constants and DAs in the LFQM.
The ground state mass spectra and the decay constants of pseudoscalar and vector heavy mesons have already been analysed by fixing the model parameters obtained from the linear and harmonic oscillator (HO) confining potentials using the 1S state HO wave function in the light-front approach [72]. Further, the mass spectra and decay constants of ground state pseudoscalar and vector light and heavy mesons have been computed in the LFQM by fixing the model parameters obtained from the linear confining potential using the two lowest order HO wave functions [73]. It is important to mention here that the fixation of model parameters has not been carried in any other potential beyond the linear and HO confining potentials. In view of this, we attempt to model the confining potential into exponential-type, which has been explored in the non-relativistic formulation [92][93][94][95][96].
The trial wave function Φ will be used for the variational principle to the QCD-motivated Hamiltonian saturating the Fock-state expansion by the constituent quark and antiquark: where the effective interaction potential V qq is the combination of (1) Coulomb plus exponential-type potential and (2) Hyperfine interaction. For our trial wave function, we find that the larger number of HO basis functions (1S, 2S, and 3S) is required to achieve the efficacy of the model calculations in contrast to the previous analyses of LFQM with the linear and HO confining potentials [72,73], the efficacy of which was already obtained with up to the two lowest order HO wave functions. It is interesting to note in this work that our LFQM predictions are comparable to each other regardless the type of confining potential as far as the efficacy of model prediction is achieved by allowing sufficient number of HO basis functions for the trial wave function. We compare the present LFQM results for the exponential-type confining potential obtained by the trial wave function composed of the three lowest HO basis functions with the previous LFQM results for the linear and HO confining potentials obtained by the trial wave function composed of up to the two lowest HO basis functions [72,73]. As in the previous work [73], the optimal model parameters have been obtained by including the hyperfine interaction term and smearing out the Dirac δ function from it in order to avoid the negative infinity problem. This study can provide important constraints on the future experiments to describe the role of variational parameters.
The present work is focused on the study of the ground state pseudoscalar and vector light (π, ρ, K, and K * ) and heavy (D, D * , D s , D * s , B, B * , B s , B * s ) mesons mass spectra. The decay constants and the twist-2 DAs of pseudoscalar, longitudinally and transversely polarized vector light and heavy mesons have been studied in detail using the LFQM.
A comparison has been made for the central potential V 0 (r) versus r for linear, HO and exponential-type potentials. The variations of decay constants of light and heavy pseudoscalar mesons as well as of the longitudinally and transversely polarized light and heavy vector mesons have also been shown in terms of the Gaussian parameter β. Using our optimized model parameters, we have computed the ground state meson mass spectra for pseudoscalar and vector light and heavy mesons. We compare the ground state mass spectra of mesons in the present work (carried out for the three lowest order HO wave functions) with that of the work in Ref. [73] (carried out for the two lowest order HO wave functions). We have computed the numerical values of decay constants as well as the ratios of pseudoscalar and vector mesons decay constants (f V /f P , f P /f P , and f V /f V ) of light and heavy mesons. The curves of normalized DAs have been plotted as a function of momentum fraction x followed by the computation of first six ξ-moments. In a nutshell, the purpose of the present work is to calculate the quark DAs of pseudoscalar, longitudinally and transversely polarized vector light and heavy mesons in the LFQM based on the idea of modelling the potential. This study will not only provide essential informations on the understanding of the universal nonperturbative quantities but also help further analyses of the hard exclusive processes.
The paper is organized as follows: In Sec. II, we begin with a brief description of the light-front framework followed by the description of our QCD-motivated Hamiltonian. In Sec. III, we discuss the procedure of fixing our model parameters through variational principle in our LFQM and present the numerical results of ground state meson mass spectra obtained from the fixed model parameters in comparison with experimental data. In Sec.
IV, we present first in Sec. IV A the formulae for the quark DAs and decay constants as well as the ξ-moments of pseudoscalar, longitudinally and transversely polarized vector mesons in the LFQM. Then, in Sec. IV B, we present the numerical results of the decay constants of pseudoscalar and vector light and heavy mesons. We also compare them with available experimental data and other theoretical model predictions. In the same subsection, we also present our results of the quark DAs for pseudoscalar, longitudinally and transversely polarized vector mesons followed by the ξ-moments. The summary and conclusions are given in Sec. V. In the Appendix, we present the analytic formula of the mass eigenvalues of the ground state pseudoscalar and vector mesons by fixing the model parameters obtained from the exponential-type potential using the mixture of three lowest order HO states as our trial wave function for the variational principle.

II. LIGHT-FRONT QUARK MODEL
We choose to work in the LFQM in which a meson bound state, consisting of a quark q and an antiquarkq with total momentum P and spin S is represented as [100,101] |M (P, S, S z ) = dp where p q(q) and λ q(q) are the on-mass shell light-front momentum and the light-front helicity of the constituent quark (antiquark), respectively. The momentump is defined as and The light-front momenta p q and pq in terms of light-front variables are defined as where x 1(2) is the longitudinal momentum fraction satisfying the relation x 1 + x 2 = 1 and k ⊥ is the relative transverse momentum of the constituent.
The momentum-space light-front wave function Ψ SSz in Eq. (1) can be expressed as a covariant form where Φ(x, k ⊥ ) describes the momentum distribution of the constituents in the bound state with x ≡ x 1 and is the invariant mass squared of the qq system. We note that M 0 is generally different from the mass M of the meson because the meson, quark and antiquark cannot be simultaneously on mass-shell. Also, the vertex factors Γ for pseudoscalar (Γ P ) and vector (Γ V ) mesons are given by withε The Dirac spinors satisfy the relations We use the radial wave function Φ(x, k ⊥ ) as an expansion of the true wave function in the three lowest order HO wave functions Φ(x, k ⊥ ) = 3 n=1 c n φ nS for both pseudoscalar and vector mesons, respectively. The 1S, 2S and 3S state HO wave functions are defined and where β represents the variational parameter and k 2 = k 2 ⊥ + k 2 z is the internal momentum of the meson. The longitudinal component k z is defined as For the variable transformation (x, k ⊥ ) → k = (k ⊥ , k z ), the Jacobian factor ∂k z /∂x is given by The meson wave function can thus be normalized as so that Our LFQM is based on the idea that we consider the radial wave function Φ(x, k ⊥ ) as a trial wave function for the variational principle to the QCD-motivated Hamiltonian saturating the Fock-state expansion by the constituent quark and antiquark. In the quark and antiquark center of mass (c.m.) frame, the meson bound system at rest is described by the following QCD-motivated effective Hamiltonian [42,100,101] where V qq is the effective interaction potential between quark and antiquark in the rest frame of the meson which is given by Coulomb (V Coul ) plus exponential-type potential (V exp ) in addition to the hyperfine interaction (V hyp ). That is, where a, b and α are the parameters of the potential, κ is the strong coupling constant which has been taken as one of the variation parameter in this work, S q ·Sq = −3/4 (1/4) for the pseudoscalar (vector) meson, respectively. We note that for the contact interaction, however, we shall smear out δ 3 (r) to avoid the negative infinity problem [73]. In

III. FIXATION OF MODEL PARAMETERS AND MESON MASS SPECTRA
We evaluate the expectation value of the system's Hamiltonian Φ|H c.m. |Φ in the variational principle using our trial wave function Φ(x, k ⊥ ) = 3 n=1 c n φ nS consisting of a variational parameter β. As we discussed in [73], when we compute Φ|H c.m. |Φ we introduce a Gaussian smearing function which weakens the singularity of δ 3 (r) in hyperfine interaction, viz., δ 3 (r) → (σ 3 /π 3/2 )e −σ 2 r 2 , to avoid the negative infinity generated by the δ function and to find the true minimum value for the mass occuring at a certain value of β. The analytic formula of mass eigenvalues for our Hamiltonian with the exponential type confining potential, i.e., M qq = Φ|H c.m. |Φ is given in the Appendix. The variational principle M qq /∂β = 0 gives us a constraint that can be used to rewrite the coupling constant κ in terms of other parameters and thus eliminate it from the mass eigenvalues.
We then assign a set of values to the externally adjustable variables (through trial and , where β P qq and β V qq denote the Gaussian parameters for pseudoscalar and vector mesons, respectively. For the exponential term of confining potential be αr , we use a typical value b = 0.385 GeV reported in Ref. [92] and find the optimum value of α from the variational principle. Following the same procedure as in Ref. [73], we use the masses of π and ρ as our input values of M qq . We fix (a, β P qq , β V qq ) for q = u, or d from the constraint that the coupling constants κ are same for all the ground state pseudoscalar and vector mesons. Effectively, we solve the following three equations: After solving Eq. (19), we obtain the values of a, β P qq , β V qq as well as the value of κ. Our obtained value κ = 0.4220 may be in contrast to κ = 0.4829 obtained from the case of linear confining potential [73]. Further, using this common value of κ, we can then calculate the β values for all the other mesons. For the best fit of the meson mass spectra, The optimized values of the constituent quark masses and other potential parameters that give the best fit of the ground state mass spectra of mesons are summarized in Table I. Due to the presence of hyperfine interaction in our variational process, we have different sets of β values for pseudoscalar and vector mesons, respectively. The optimal Gaussian parameters β P qq and β V qq for pseudoscalar and vector mesons obtained by the variational principle are listed in Table II.   Table III, comparing with the experimental data and the previous results obtained from the linear and HO potentials [72,73]. The predictions for the ground state meson masses in our LFQM obtained from the exponential-type confining potential and smeared hyperfine interaction are in a resonable agreement with the experimental data [109]. We also note in Table III that our predictions are consistent with the ones obtained from the linear [72,73] and HO [72] potential models.

IV. QUARK DISTRIBUTION AMPLITUDES AND DECAY CONSTANTS
Having fixed all the model parameters for the present analysis achieving a reasonable agreement with the data for the meson mass spectra, we now apply the present model calculation to the wave function related observables such as the quark DAs and decay constants in this section. We first summarize the relevant formulae in Sec. IV A and TABLE III. Ground state mass spectra (in units of GeV) of pseudoscalar (K, D (s) , B (s) ) and vector (K * , D * (s) , B * (s) ) mesons obtained from the exponential type potential and their comparison with the experimental data [109] and the LFQM results obtained from the linear and HO potentials [72,73]. A. Summary of Formulae The quark DAs are defined in terms of the matrix elements of non-local operators that are sandwiched between the vacuum and the meson states [41] 0|q(0)γ µ γ 5 q(0)|P (P ) = if P P µ Here φ P , φ V and φ V ⊥ are the twist-2 DAs of pseudoscalar, longitudinally and transversely polarized vector mesons, respectively. The explicit forms of quark DAs in our LFQM are given by [45] φ P (x) = 2 From Eq. (26), we can define the decay constants for the pseudoscalar and the vector mesons as We may also define the expectation value of the longitudinal momentum, so-called ξmoments, as follows 2x).

B. Numerical Results
In order to perform numerical calculations of decay constants for pseudoscalar and vector mesons, we use the model parameters given in Table I and Table II In Table IV, we present our predictions for the decay constants of pseudoscalar, longitudinally and transversely polarized vector light (π, ρ, K, and K * ) mesons obtained using the parameters given in Tables I and II and compare them with the LFQM [45,73], BS model [86], RQM [91], SR [20], and lattice QCD [59] predictions as well as with the available experimental data [109]. We note that our predictions for the pseudoscalar decay predictions are also consistent with the other theoretical model results. The values of ratios f ρ /f π , f K * /f K , f K /f π , and f K * /f ρ for light mesons are listed in Table V so as to have a deeper comprehension of the quantitative difference between pseudoscalar and vector mesons decay constants (f V /f P , f P /f P , and f V /f V ). The comparison has also been made with the available experimental data [109] and other theoretical model calculations [20,45,59,73,86,91]. It is evident that our results for the ratios f ρ /f π = 1.88, f Ds /f D and f D * s /f D * and compared them with the available experimental data and other theoretical calculations. We can see that our result for the ratio f Ds /f D = 1.11 is comparable with the available experimental data f Ds /f D = 1.27 ± 0.03 [109]. We can also see that the ratios f Ds /f D = 1.11 and f D * s /f D * = 1.10 in this work are in good agreement with f Ds /f D = 1.11 and f D * s /f D * = 1.13 from the previous LFQM [73], f Ds /f D = 1.09 from LFHQCD [77] and f Ds /f D = 1.10 ± 0.02 and f D * s /f D * = 1.11 ± 0.03 from the lattice QCD [83]. Also from Table VII, we can observe that our results for the ratios f D * /f D = 1.17 and f D * s /f Ds = 1.16 are also comparable with the other theoretical model calculations. In Table VIII, we present our predictions for the decay constants of pseudoscalar, longitudinally and transversely polarized vector B mesons and compare them with the LFQM [41,[72][73][74], LFHQCD [77,78], SR [79][80][81], lattice QCD [82][83][84], BS model [86][87][88], RQM [89][90][91] and NRQM [92,[94][95][96] predictions as well as with the available experimental data [109]. One can note that our prediction for the decay constant We should note that in the case of heavy mesons, the ratios f V /f P of the vector and the pseudoscalar mesons are larger for the case of D (s) mesons than for the case of B (s) mesons as one may expect from the heavy quark symmetry. This may also be accounted from the last terms of Eqs. (24) and (25)  appears relatively more pronounced in D s and D * s than in B s and B * s , which may also be understood from the heavy quark symmetry. To further exploit such relative effects, we present in Table X and XI the first six ξ-moments for D and B mesons, respectively.
It is found from the tables that the ξ-moments of B mesons are higher in magnitude as compared to the D mesons.

V. SUMMARY AND CONCLUSIONS
In this work, we have studied the mass spectra, decay constants and the twist-2 DAs of pseudoscalar and vector light (π, K) and heavy (D, D s , B, B s ) mesons within the LFQM with the exponential-type confining potential. Our LFQM is constrained by the variational principle for the QCD-motivated effective Hamiltonian not only with the confining potential but also with the Coulomb and hyperfine interaction. We have used a Gaussian smearing function to weaken the singularity of the Dirac δ function in hyperfine interaction so as to avoid the issue of negative infinity [73]. We have calculated the mass spectra of the ground state pseudoscalar and vector light and heavy mesons as well as the decay constants of the corresponding mesons using the mixed wave function Φ of 1S, 2S and 3S HO states as the trial wave function. We also compared our results with available experimental data and the previous LFQM results with the linear and HO potentials [72,73] as well as other theoretical model predictions. We note that our LFQM predictions are Even though the observables that we have estimated have been previously calculated in the light-front framework, the present work is based on the idea of modelling the potential in a rather significantly different way from the previous works [72,73]. We anticipate to study further other wave function related observables such as meson transition form factors and also analyse radially excited meson states using the larger number of HO basis functions.