2S and 3S State Masses and decay constants of heavy-flavour mesons in a non-relativistic QCD potential model with three-loop effects in V-scheme

We make an analysis of three -loop effects of the strong coupling constant in the study of masses and decay constants of the heavy-flavour pseudo-scalar mesons (PSM) D, Ds\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_s$$\end{document} , B, Bs,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_s,$$\end{document}Bc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_c$$\end{document}, ηc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _c$$\end{document} and ηb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _b$$\end{document} in a non-relativistic QCD potential model using Dalgarno’s perturbation theory (DPT). The first order mesonic wavefunction is obtained using Dalgarno’s perturbation theory. The three-loop effects of strong coupling constant are included in the wave function in co-ordinate space and then used to examine the static and dynamic properties of the heavy-flavour mesons for 2S and 3S higher states. The results are compared with the other models available and are found to be compatible with available experimental values. In V-scheme, the three-loop effects on masses and decay constants of heavy-flavour mesons show a significant result.


Introduction
The non-relativistic predictions of Potential Models with a non-relativistic Hamiltonian for the heavy-light and heavyheavy mesons are found to be in fair agreement with the updated experimental data, theoretical results like QCD sum rule [1,2], Lattice results [3] and relativistic harmonic confinement model (RHCM) [53]. The static potential between the two heavy quarks is a fundamental quantity in QCD [4]. While its one loop corrections are computed in [5,6], the corresponding two-loop effects were reported in late 1990's [7][8][9]. Numerical results are obtained first for fermionic contributions [10][11][12], whereas the analytical results are more recent [12]. Some important hadronic properties are the pseudoscalar meson mass M P and decay constant f P . Phenomenological study of two-loop effects in the static and a e-mail: rashidul999@gmail.com (corresponding author) dynamic properties of heavy-flavour mesons using a linear cum Coulomb Cornell potential has been reported in the recent years [14,15]. In this work, quantum perturbation approach [16,17] is used to calculate the approximate analytical forms of heavy flavored mesons. Here specifically the linear part of the potential V (r ) = − 4α s 3r + br is used as perturbation.
The most common perturbative method is Dalgarno's perturbation theory (DPT) [21,[23][24][25], which is a stationary static perturbation theory. The non-relativistic potential model has been found successful for heavy-heavy B, η c and η b families. The study of the wave functions of heavy-flavour mesons like B and D and η are important both analytically and numerically for studying the properties of strong interaction between heavy-light and heavy-heavy quarks as well as for investigating the mechanism of heavy meson decays. In this work, we have obtained a total first order corrected wavefunction for 2S and 3S states using Dalgarno's method of perturbation [30] with linear part of the Cornell potential [19,31] as perturbation in co-ordinate space. This wavefunction is then used to estimate the masses and decay constants of heavy-light and heavy-heavy pseudo-scalar mesons in this improved QCD Potential model approach.
One aim of the present work is to make an analysis of the contribution of three-loop effects in the improved strong coupling constant α V ( 1 r ) in V -scheme, which in turn contributes to the spin-spin interaction [27][28][29] term present in the expression of mass and decay constant of the heavy-flavour meson. In addition, the non-relativistic binding energy effect between the two quark-antiquark composition of the heavyflavour meson is newly incorporated in the expression of PSM mass, which was absent in our some previous works [23][24][25]37,39,41].
The rest of the paper is organised as follows: Sect. 2 contains formalism, Sect. 3 contains results while Sect. 4 includes the conclusion.

Formalism
2.1 V-scheme: three-loop effects V-scheme is a standard way of taking into account the higher order effects of QCD, which are expressed as power series in the running strong coupling constant α M S in M S-scheme. The two-loop static potential in V-scheme which is also used as the three-loop static potential defined as [8,33,34], Here, α V is the effective strong coupling constant and C F is the color factor, given as, where N C is the no. of colors. Generally, the quark-gluon interaction is characterised by strong coupling constant α M S (q 2 ) in a dimensionally independent M S-scheme [7,20]. As discussed in the introduction , three-loop effects have been reported numerically first for fermionic contribution [10] followed by the gluonic counter part [10][11][12]. Analytic three-loop static potential have been discussed only in the year 2016 [12]. These effects are invariably reported in momentum space where the author sets q 2 = μ 2 as the renormalization scale to suppress the infrared logarithmics. For the three-loop effects, we follow [11] the numerical solution for the potential at three-loop level in momentum space is given by, The corresponding expression in co-ordinate space will be, Using Eq. (1), we obtain the relationship between the improved strong coupling constant α V ( 1 r ) and the standard leading order strong coupling constant in M S-scheme at N 3 L O level is given by: From the above equation, it is observed that at one-, twoand three-loop order, a large screening of the nonfermionic contributions by the n f terms which is most prominent in the case of a 3 for n f = 5.

2S state wave-function of the heavy-flavour mesons
For shell L, we take n = 2 and l = 0 ; the 2S state normalized wave-function [42] is given by: In Dalgarno's perturbation theory, we make a small deformation to the Hamiltonian of the system as , where H 0 is the Hamiltonian of the unperturbed system and H is the perturbed Hamiltonian. The approximation method is most suitable when H is close to the unperturbed Hamiltonian H 0 , i.e. H is small. The standard potential is [19], This Coulomb-plus-linear potential, called Cornell potential is an important ingredient of the model which is established on the two kinds of asymptotic behaviours: ultraviolet at short distance (Coulomb like) and infrared at large distance (linear confinement term).
The Schrödinger equation takes the form as, so that the first-order perturbed eigenfunction ψ (1) and eigen energy W (1) can be obtained using the relation, where H 0 and H are the parent and perturbed Hamiltonian defined as, and With Cornell potential we get two choices in DPT: 1. Coulomb Parent, Linear Perturbation (CP). 2. Linear Parent, Coulomb Perturbation (LP).
In our present analysis we consider the first option. The radial Schrodinger equation for l = 0 and n = 2 is, where ψ (0) 20 is unperturbed wave function and is defined in Eq. (5) and ψ (1) 20 is the first order correction to the wave function.
To solve the above equation, let us start from [24], With this substitution, Eq. (15) takes the form as: with, Putting as in [24], Equation (17) becomes, The method of Frobenius [56] is a power series solution. Considering the only four terms in the series and neglecting the higher order terms for 2S state, with this the final form of the Eq. (22) is obtained as, The expectation energy (eigen energy) is easily obtained using mathematica-9 as , Equating co-efficients of r −1 ,r 0 r and r 2 on both sides of Eq. (24), we get the values of constants as: Hence Eq. (16) becomes, Hence, using Dalgarno's perturbation theory with Coulombic parent, we get the total wave-function of the form corrected up to first order, where the normalization constant is, where, a 0 = 1 μA ; the value of b is b = 0.183 GeV 2 is the confinement parameter [5,9]. μ = m 1 m 2 m 1 +m 2 is the reduced mass.

3S state wave-function of the heavy-flavour mesons
For shell M, we take n = 3 and l = 0 ; the 3S state normalized wave-function [42] is given by, represents the unperturbed wave function for 3S state. For l = 0 and n = 3, the corresponding Eq. (15) for 3S state becomes, where ψ (1) 30 is the first order correction to the wave function. Similarly, as in Eq. (16) Let, with this substitution the Eq. (37) takes the form as, with, The corresponding substitution as in Eq. (21); Therefore, Eq. (39) is obtained as, Similar way, the corresponding only four terms in the method of Frobenius [56] for 3S state, With this, the final form of the Eq. (42) is obtained as, The expectation energy is obtained using mathematica-9 as , Equating the co-efficients of r −1 , r , r 2 and r 3 , we get the values of the constants as: Hence, Eq. (38) yields, Therefore, the total wave-function corrected up to first order using Dalgarno's perturbation theory is, with, where the normalization constant is obtained from,

Wave function at origin (WFO)
At origin, r = 0; WFO for 2S state (Eq n .31) is given by, Similarly, WFO, for 3S state (Eq n .51) is given by,

The expression of mass and decay constant of pseudo-scalar meson
Fermi-Breit Hamiltonian: We take the non-relativistic two body Schrodinger equation (8) viz., Where H 0 is the free Hamiltonian for two quarks of masses m i and m j and three momenta P i and P j . H 0 is defined as, and H is the Fermi-Breit Hamiltonian with confinement which is defined as [51,55], Here, δ H Con f δr Here, S i and S j are the spins of the i th and j th quarks separated by a distance r . For ground state (l = 0), only the contact term proportional to δ 3 (r ) contributes and the Hamiltonian takes the simpler form as: In the present work, c sets to be zero. To compute mass of the pseudoscalar mesons the spin-spin interaction possessing the form given by [49,50], The mass and the decay constant of heavy-flavour pseudoscalar meson including only spin-spin interaction are given in [26,27] as: where, m 1 and m 2 are the masses of quark-antiquark and α s is the strong coupling constant identified as α V in the present work, E n,l is the non-relativistic binding energy between the quark-antiquark composition and the van Royen-Weisskopf formula [54] for the decay of pseudo-scalar meson is, Here, α V is the improved strong coupling constant in Vscheme defined by Eq. (4) and |ψ(0)| is the wave function at origin (WFO).

Input parameters used in the calculation
With the formalism developed in Sec.2, we calculate the masses and decay constants of pseudo-scalar heavy-flavour mesons using Eqs. We make a comprehensive comparison of our results with QCD sum rule [1,2], Lattice results [3,19,43,44] other models like [24], RHCM [53] and the recent experimental values [36].
We use the usual expression for strong coupling constant in M S scheme for lowest order (LO level) is given by [40] Here, m Q is the mass of the heavy quark and QC D is the QCD scale parameter having values 0.292 GeV and 0.210 GeV for n f = 4 and n f = 5 respectively.

Calculation of effective strong coupling constant
Using Eq. (4), we tabulate the values of α V ( 1 r ) taking into account one-loop (NLO), two-loop (N 2 L O) and three-loop Table 1 for n f = 4 and n f = 5. It shows that for n f = 4, the enhancements are respectively 21%, 48% and 62% while for n f = 5, the corresponding enhancements are 8% , 14% and 17% respectively. The anti-screening effects of gluons seem to play an important role for n f = 5.  Table 2 below. It is seen that from the above Table 2, the magnitude of the non-relativistic binding energy is always greater for 2S state than 3S state, which indicates, the non-relativistic binding energy decreases with increasing higher states.

Calculation of parameter A
Using Eq. (20) with the values of α V from Table 1, we calculate the parameter A for the same heavy-flavour pseudoscalar mesons given in the Table 3 below.  Table 5.     Tables 6 and 7 respectively with the comparison of results of the other models like QCD sum rule [1,2], Lattice results [3,19,43,44], relativistic harmonic potential model (RHPM) [53] and experimental data [36,[45][46][47].  Tables 8 and 9. Comparative analysis of pseudo-scalar meson mass spectra from Tables 6 and 8 provides a well agreement of our results in both 2S and 3S states while the decay constants from Tables 7 and 9 are found to be in better harmony only for 2S state with experimental values and other models available.

Sources of uncertainties and possible future improvements
Let us discuss the sources of uncertainties in the calculation and the possible future improvements. The sources of uncertainties arise mainly:  Let us now discuss about the possible future improvements and their applications: In the present work we have not introduced the relativistic effect in the light quark. For the future improvement the light quark can be considered relativistically with the Hamiltonian H = M+ p 2 2m + p 2 + m 2 +V (r ), where M is the mass of heavy quark and m is the mass of light quark as in [33,37,57]. Similarly, the QCD correction factor C 2 (α V ) [37,58] can also be introduced in van Royen-Weisskopf formula [37,54] for the decay of pseudo-scalar meson. The improved model then can be applied in the study of Branching ratio, Oscillation frequency, Leptonic decay as well [37].

Conclusion
The spectroscopic properties of heavy-light and extended heavy-heavy favour system mesons are studied as a potential scheme with Dalgarno's Perturbation Theory.  [37,58] and relativistic effects at least minimally [37] are expected to improve the results.
In conclusion, the Dalgarno's Perturbation approach with the option of linear perturbation and three-loop effects employed here is found to be successful in the study of heavyflavour mesons with a Coulombic plus linear potential. More-over, our results suggest, the treatment of light-flavour relativistically and heavy-flavour non-relativistically seems to be appropriate in light of the successful predictions of the various properties of heavy-light and heavy-heavy systems. The parameters and the results obtained here can be useful in the study of the leptonic and semi-leptonic decays of these mesons.
Acknowledgements One of the authors (R. Hoque) acknowledges Maulana Azad National Fellowship, India for financial support by providing Fellowship during the research work. He also thanks the Head of the department of Physics, Gauhati University and the Head of the department of Physics, Pandu College, Guwahati, India for providing the necessary facilities.

Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors' comment: Our research work is basically to build a model to estimate static and dynamic properties of heavy-flavour mesons within phenomenological Quantum Chromodynamics. So, data deposition is not necessarily emphasized in our present 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 copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/. Funded by SCOAP 3 .