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

Abstract

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, $D_s$ , B, $B_s,$ $B_c$, $\eta _c$ and $\eta _b$ 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.

1 Introduction

The non-relativistic predictions of Potential Models with a non-relativistic Hamiltonian for the heavy-light and heavy-heavy 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 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 $V(r)=-\frac{4\alpha _s}{3r} + b r$ 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, $\eta _c$ and $\eta _b$ families. The study of the wave functions of heavy-flavour mesons like B and D and $\eta$ 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 $\alpha _V(\frac{1}{r})$ 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 heavy-flavour 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.

2 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 $\alpha _{{\overline{MS}}}$ in ${\overline{MS}}$-scheme. The two-loop static potential in V-scheme which is also used as the three-loop static potential defined as [8, 33, 34],

$$\begin{aligned} V(r)=-\frac{C_F\alpha _ V\left( \frac{1}{r}\right) }{r} \end{aligned}$$

(1)

Here, $\alpha _V$ is the effective strong coupling constant and $C_F$ is the color factor, given as, $C_F=\frac{(N_C^2-1)}{2N_C}$, where $N_C$ is the no. of colors. Generally, the quark–gluon interaction is characterised by strong coupling constant $\alpha _{{\overline{MS}}}(q^2)$ in a dimensionally independent ${\overline{MS}}$-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 $q^2=\mu ^{'2}$ 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,

$$\begin{aligned} V(|q|)= & {} -\frac{4\pi C_F \alpha _s(|\overline{q^2}|)}{\overline{q^2}} \nonumber \\&\times \left[ 1+\left( \frac{\alpha _s}{\pi }\right) \left( 2.5833-0.2778 n_f\right) \right. \nonumber \\&+\left( \frac{\alpha _s}{\pi }\right) ^2\left( 28.5468-4.1471 n_f+0.0772n_f^{2}\right) \nonumber \\&+\left( \frac{\alpha _s}{\pi }\right) ^3 \left( 209.884-51.4048 n_f \right. \nonumber \\&\left. \left. +2.9061 n_f^{2}-0.0214 n_f^{3}\right) \right] \end{aligned}$$

(2)

The corresponding expression in co-ordinate space will be,

$$\begin{aligned} V(r)= & {} -\frac{C_F}{r} \alpha _s(\mu ^{'2}) \left[ 1+\left( \frac{\alpha _s}{\pi }\right) (2.5833-0.2778 n_f) \right. \nonumber \\&+\left( \frac{\alpha _s}{\pi }\right) ^2(28.5468-4.1471 n_f+0.0772n_f^{2}) \nonumber \\&+ \left( \frac{\alpha _s}{\pi }\right) ^3 (209.884-51.4048 n_f \nonumber \\&\left. +2.9061 n_f^{2}-0.0214 n_f^{3}) \right] \end{aligned}$$

(3)

Using Eq. (1), we obtain the relationship between the improved strong coupling constant $\alpha _V(\frac{1}{r})$ and the standard leading order strong coupling constant in ${\overline{MS}}$-scheme at $N^3LO$ level is given by:

$$\begin{aligned} \alpha _V\left( \frac{1}{r}\right)= & {} \alpha _s(\mu ^{'2}) \left[ 1+\left( \frac{\alpha _s}{\pi }\right) (2.5833-0.2778 n_f) \right. \nonumber \\&+\left( \frac{\alpha _s}{\pi }\right) ^2 (28.5468-4.1471 n_f+0.0772n_f^{2}) \nonumber \\&+ \left( \frac{\alpha _s}{\pi }\right) ^3 (209.884-51.4048 n_f \nonumber \\&\left. +2.9061 n_f^{2}-0.0214 n_f^{3})\right] \end{aligned}$$

(4)

From the above equation, it is observed that at one-, two- and 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 $n_f = 5$.

2.2 Dalgarno’s perturbation theory

2.2.1 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:

$$\begin{aligned} \psi ^{(0)}_{20}(r)={\left( \frac{1}{2\sqrt{2\pi }}\right) \left( \frac{1}{a_0}\right) ^{\frac{3}{2}}} \left( 1-\frac{r}{2a_0}\right) e^{-\frac{r}{2a_0}} \end{aligned}$$

(5)

In Dalgarno’s perturbation theory, we make a small deformation to the Hamiltonian of the system as ,

$$\begin{aligned} H=H_0+H^\prime \end{aligned}$$

(6)

where $H_0$ is the Hamiltonian of the unperturbed system and $H^\prime$ is the perturbed Hamiltonian. The approximation method is most suitable when H is close to the unperturbed Hamiltonian $H_0$, i.e. $H^\prime$ is small. The standard potential is [19],

$$\begin{aligned} V(r)=-\frac{4\alpha _V}{3r}+br. \end{aligned}$$

(7)

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,

$$\begin{aligned} H|\psi \rangle =(H_0+H^\prime )|\psi \rangle =E|\psi \rangle \end{aligned}$$

(8)

so that the first-order perturbed eigenfunction $\psi ^{(1)}$ and eigen energy $W^{(1)}$ can be obtained using the relation,

$$\begin{aligned} H_0 \psi ^{(1)} + H^\prime \psi ^{(0)}=W^{(0)}\psi ^{(1)} + W^{(1)} \psi ^{(0)} \end{aligned}$$

(9)

where $H_0$ and $H^{'}$ are the parent and perturbed Hamiltonian defined as,

$$\begin{aligned} H_0= & {} \frac{-\bigtriangledown ^2}{2\mu }-\frac{A}{r} \end{aligned}$$

(10)

$$\begin{aligned} H^{'}= & {} br \end{aligned}$$

(11)

and

$$\begin{aligned} W^{(0)}= & {} \langle \psi ^{(0)}\vert H_0 \vert \psi ^{(0)} \rangle =-\frac{\mu A^{2}}{2} \end{aligned}$$

(12)

$$\begin{aligned} W^{(1)}= & {} \langle \psi ^{(0)}\vert H^\prime \vert \psi ^{(0)} \rangle \end{aligned}$$

(13)

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,

$$\begin{aligned}&\left[ -\frac{1}{2\mu } \left( \frac{d^2}{dr^2}+\frac{2}{r} \frac{d}{dr}\right) +\frac{l(l+1)}{2}-\frac{A}{r}-W^{(0)}\right] \psi _{20}^{(1)} \nonumber \\&\quad =-\left[ br-W^{(1)}\right] \psi _{20}^{(0)} \end{aligned}$$

(14)

$$\begin{aligned}&\left[ \frac{d^2}{dr^2}+\frac{2}{r}\frac{d}{dr}+\frac{2\mu A}{r}-\mu ^2A^2 \right] \psi _{20}^{(1)} \nonumber \\&\quad =2\mu \left[ br-W^{(1)}\right] \psi _{20}^{(0)} \end{aligned}$$

(15)

where $\psi _{20}^{(0)}$ is unperturbed wave function and is defined in Eq. (5) and $\psi _{20}^{(1)}$ is the first order correction to the wave function.

To solve the above equation, let us start from [24],

$$\begin{aligned} \psi _{20}^{(1)}=b r R(r) \end{aligned}$$

(16)

With this substitution, Eq. (15) takes the form as:

$$\begin{aligned}&(br)\frac{d^2R(r)}{dr^2}+4b\frac{dR(r)}{dr}+\frac{2bR(r)}{r} +\frac{2bR(r)}{a_0}-\frac{brR(r)}{a_0^{2}} \nonumber \\&\quad =K (br-W^{(1)}) \left( 1-\frac{r}{2a_0}\right) e^{-\frac{r}{2a_0}} \end{aligned}$$

(17)

with,

$$\begin{aligned} K= & {} (2\mu )\frac{1}{2\sqrt{2\pi }}\left( \frac{1}{a_0} \right) ^{\frac{3}{2}} \end{aligned}$$

(18)

$$\begin{aligned} a_0= & {} \frac{1}{\mu A} \end{aligned}$$

(19)

$$\begin{aligned} A= & {} \frac{4\alpha _V}{3} \end{aligned}$$

(20)

Putting as in [24],

$$\begin{aligned} R(r)=F(r) e^{-\frac{r}{2a_0}} \end{aligned}$$

(21)

Equation (17) becomes,

$$\begin{aligned}&(br)F^{''}(r)+\left( 4b-\frac{br}{a_0}\right) F^{'}(r) +\left( \frac{2b}{r}-\frac{3br}{4a_0^{2}}\right) F(r) \nonumber \\&\quad =K (br-W^{(1)})\left( 1-\frac{r}{2a_0}\right) \end{aligned}$$

(22)

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,

$$\begin{aligned} F(r)=\frac{A_{-1}}{r}+A_0+A_1r+A_2r^2 \end{aligned}$$

(23)

with this the final form of the Eq. (22) is obtained as,

$$\begin{aligned}&\left( 2ba_0+\frac{bA_{-1}}{a_0}\right) \frac{1}{r} \left( 6bA_1-\frac{3bA_{-1}}{4a_0^{2}}\right) r^{0} \nonumber \\&\quad +\left( 12bA_2-\frac{bA_1}{a_0}-\frac{3bA_0}{4a_0^{2}}\right) r -\left( \frac{2bA_2}{a_0}+\frac{3bA_1}{4a_0^{2}}\right) r^{2} \nonumber \\&\quad -\left( \frac{3bA_2}{4a_0^{2}}\right) r^3=-(KW^{1})r^0 +\left( Kb+\frac{KW^{1}}{2a_0}\right) r \nonumber \\&\quad -\left( \frac{Kb}{2a_0}\right) r^2 \end{aligned}$$

(24)

The expectation energy (eigen energy) is easily obtained using mathematica-9 as ,

$$\begin{aligned} \langle W^{(1)} \rangle = \int _0^{\infty } 4 \pi r^2\psi ^{0}H^{'}\psi ^{0} = 6ba_0 \end{aligned}$$

(25)

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:

$$\begin{aligned} A_{-1}= & {} \frac{40Ka_0^{3}}{3} \end{aligned}$$

(26)

$$\begin{aligned} A_{0}= & {} \frac{20Ka_0^{2}}{3} \end{aligned}$$

(27)

$$\begin{aligned} A_{1}= & {} \frac{2Ka_0}{3} \end{aligned}$$

(28)

$$\begin{aligned} A_2= & {} 0 \end{aligned}$$

(29)

Hence Eq. (16) becomes,

$$\begin{aligned} \psi _{20}^{(1)}=K(K_1-K_2r+K_3r^2) e^{-\frac{r}{2a_0}} \end{aligned}$$

(30)

with, $K_1=\frac{40 b a_0^{3}}{3}$, $K_2=\frac{20 b a_0^{2}}{3}$, $K_3=\frac{2 b a_0}{3}$.

Hence, using Dalgarno’s perturbation theory with Coulombic parent, we get the total wave-function of the form corrected up to first order,

$$\begin{aligned} \psi ^{total}_{20}(r)= & {} \psi ^{(0)}_{20}(r)+\psi ^{(1)}_{20}(r) \nonumber \\= & {} N\left[ P-Qr+Rr^2\right] e^{-\frac{r}{2a_0}}, \end{aligned}$$

(31)

$$\begin{aligned} P= & {} K\left[ \frac{1}{2\mu }+K_1\right] \end{aligned}$$

(32)

$$\begin{aligned} Q= & {} K \left[ \frac{1}{4\mu a_0}+K_2\right] \end{aligned}$$

(33)

$$\begin{aligned} R= & {} KK_3 \end{aligned}$$

(34)

where the normalization constant is,

$$\begin{aligned} N=\frac{1}{\left[ \int _0^{\infty } 4 \pi r^2\left[ P-Qr+Rr^2\right] ^2e^{-\frac{r}{a_0}}dr\right] ^{\frac{1}{2}}}. \end{aligned}$$

(35)

where, $a_{0}=\frac{1}{\mu A}$ ; the value of b is $b =0.183$ GeV$^2$ is the confinement parameter [5, 9]. $\mu =\frac{m_1m_2}{m_1+m_2}$ is the reduced mass.

2.2.2 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,

$$\begin{aligned} \psi _{30}^{(0)}(r)={\left( \frac{1}{3\sqrt{3\pi }}\right) \left( \frac{1}{a_0}\right) ^{\frac{3}{2}}} \left( 1-\frac{2r}{2a_0}+\frac{2r^2}{27a_0^3}\right) e^{-\frac{r}{3a_0}}\nonumber \\ \end{aligned}$$

(36)

represents the unperturbed wave function for 3S state. For $l=0$ and $n=3$, the corresponding Eq. (15) for 3S state becomes,

$$\begin{aligned}&\left[ \frac{d^2}{dr^2}+\frac{2}{r}\frac{d}{dr}+\frac{2\mu A}{r}-\mu ^2A^2 \right] \psi _{30}^{(1)} \nonumber \\&\quad =2\mu \left[ br-W^{(1)}\right] \psi _{30}^{(0)} \end{aligned}$$

(37)

where $\psi _{30}^{(1)}$ is the first order correction to the wave function.

Similarly, as in Eq. (16) Let,

$$\begin{aligned} \psi _{30}^{(1)}=b r R(r) \end{aligned}$$

(38)

with this substitution the Eq. (37) takes the form as,

$$\begin{aligned}&(br)\frac{d^2R(r)}{dr^2}+4b\frac{dR(r)}{dr}+\frac{2bR(r)}{r} +\frac{2bR(r)}{a_0}-\frac{brR(r)}{a_0^{2}} \nonumber \\&\quad = L (br-W^{(1)}) \left( 1-\frac{2r}{3a_0}+\frac{2r^2}{27a_0^3}\right) e^{-\frac{r}{3a_0}} \end{aligned}$$

(39)

with,

$$\begin{aligned} L=(2\mu )\frac{1}{3\sqrt{3\pi }}\left( \frac{1}{a_0}\right) ^{\frac{3}{2}} \end{aligned}$$

(40)

The corresponding substitution as in Eq. (21);

$$\begin{aligned} R(r)=F(r) e^{-\frac{r}{3a_0}} \end{aligned}$$

(41)

Therefore, Eq. (39) is obtained as,

$$\begin{aligned}&(br)F^{''}(r)+ \left( 4b-\frac{2br}{3a_0}\right) F^{'}(r) +\left( \frac{2b}{3a_0}-\frac{8br}{9a_0^{2}}+\frac{2b}{r}\right) F(r) \nonumber \\&\quad =L(br-W^{(1)})\left( 1-\frac{2r}{3a_0}+\frac{2r^2}{27a_0^{2}}\right) \end{aligned}$$

(42)

Similar way, the corresponding only four terms in the method of Frobenius [56] for 3S state,

$$\begin{aligned} F(r)=\frac{A^{'}_{-1}}{r}+A^{'}_0+A^{'}_1r+A^{'}_2r^2 \end{aligned}$$

(43)

With this, the final form of the Eq. (42) is obtained as,

$$\begin{aligned}&\left( 2bA^{'}_0+\frac{4bA^{'}_{-1}}{3a_0}\right) \frac{1}{r} \nonumber \\&\qquad +\left( 6bA^{'}_{1}+\frac{2bA^{'}_0}{3a_0}-\frac{8bA^{'}_{-1}}{9a_0^2}\right) r^0 \nonumber \\&\qquad +\left( 12bA^{'}_2-\frac{8bA^{'}_0}{9a_0^2}\right) r -\left( \frac{2bA^{'}_2}{3a_0}+\frac{8bA^{'}_1}{9a_0^2}\right) r^2\nonumber \\&\qquad -\left( \frac{8bA^{'}_2}{9a_0^2}\right) r^3 \nonumber \\&\quad = L\left[ br-\left( \frac{2b}{3a_0}\right) r^2 +\left( \frac{2b}{27a_0^2}\right) r^3-W^{(1)} \right. \nonumber \\&\qquad \quad \left. +\left( \frac{2r}{3a_0} W^{(1)}\right) r-\left( \frac{2r^2}{27a_0^2}W^{(1)}\right) r^2\right] \end{aligned}$$

(44)

The expectation energy is obtained using mathematica-9 as ,

$$\begin{aligned} \langle W^{(1)} \rangle = \frac{4b}{a_0^2} \end{aligned}$$

(45)

Equating the co-efficients of $r^{-1}$, r, $r^2$ and $r^3$, we get the values of the constants as:

$$\begin{aligned} A^{'}_{-1}= & {} \frac{27La_0^{3}}{8}+\frac{9L}{2} \end{aligned}$$

(46)

$$\begin{aligned} A^{'}_{0}= & {} - \left( \frac{9La_0^{2}}{4}+\frac{3L}{a_0} \right) \end{aligned}$$

(47)

$$\begin{aligned} A^{'}_{1}= & {} \frac{39La_0}{48}+\frac{L}{3a_0^2} \end{aligned}$$

(48)

$$\begin{aligned} A^{'}_2= & {} -\frac{L}{12} \end{aligned}$$

(49)

Hence, Eq. (38) yields,

$$\begin{aligned} \psi _{30}^{(1)}=b(A^{'}_1+A^{'}_0r+A^{'}_1r^2+A^{'}_2r^3) e^{-\frac{r}{3a_0}} \end{aligned}$$

(50)

Therefore, the total wave-function corrected up to first order using Dalgarno’s perturbation theory is,

$$\begin{aligned} \psi ^{total}_{30}(r)= & {} \psi ^{(0)}_{30}(r)+\psi ^{(1)}_{30}(r)\nonumber \\= & {} N^{'}\left[ L_1+L_2r+L_3r^2+L_4r^3\right] e^{-\frac{r}{3a_0}}, \end{aligned}$$

(51)

with,

$$\begin{aligned} K^{'}= & {} \frac{1}{3\sqrt{3\pi }}\left( \frac{1}{a_0}\right) ^{\frac{3}{2}} \end{aligned}$$

(52)

$$\begin{aligned} L_1= & {} \left[ K^{'}+bA^{'}_{-1}\right] \end{aligned}$$

(53)

$$\begin{aligned} L_2= & {} \left[ bA^{'}_0-\frac{2K^{'}}{3a_0}\right] \end{aligned}$$

(54)

$$\begin{aligned} L_3= & {} \left[ bA^{'}_1+\frac{2K^{'}}{27a_0^2}\right] \end{aligned}$$

(55)

$$\begin{aligned} L_4= & {} \left[ bA^{'}_2\right] \end{aligned}$$

(56)

where the normalization constant is obtained from,

$$\begin{aligned} N^{'}=\frac{1}{\left[ \int _0^{\infty } 4 \pi r^2\left[ L_1+L_2r+L_3r^2+L_4r^3\right] ^2e^{-\frac{2r}{3a_0}}dr\right] ^{\frac{1}{2}}}. \nonumber \\ \end{aligned}$$

(57)

2.3 Wave function at origin (WFO)

At origin, r = 0; WFO for 2S state ($Eq^n. 31$) is given by,

$$\begin{aligned} \psi ^{total}_{20}(0)=NP=NK\left( \frac{1}{2\mu }+\frac{40ba_0^3}{3}\right) \end{aligned}$$

(58)

Similarly, WFO, for 3S state ($Eq^n. 51$) is given by,

$$\begin{aligned} \psi ^{total}_{30}(0)=N^{'}L_1=N^{'}\left[ K^{'}+b\left( \frac{27La_0^3}{8}+\frac{9L}{2}\right) \right] \end{aligned}$$

(59)

2.4 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.,

$$\begin{aligned} H|\psi \rangle =(H_0+H^\prime )|\psi \rangle =E|\psi \rangle , \end{aligned}$$

(60)

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,

$$\begin{aligned} H_0 = \frac{P_{i}^{2}}{2m_i} + \frac{P_{j}^{2}}{2m_j} \end{aligned}$$

(61)

and H is the Fermi–Breit Hamiltonian with confinement which is defined as [51, 55],

$$\begin{aligned} H(r) = H^{Conf}(r) + H^{hyp}(r) + H^{S.O.}(r) \end{aligned}$$

(62)

Here,

$$\begin{aligned} H^{Conf}(r)= & {} \left( -\frac{\alpha _s(r)}{r} + \frac{3 br}{4} + \frac{3 c}{4}\right) \left( \bar{ F_i}.\bar{F_j} \right) \end{aligned}$$

(63)

$$\begin{aligned} H^{hyp}(r)= & {} -\frac{\alpha _s(r)}{m_i m_j} \nonumber \\&\times \left[ \frac{8\pi }{3} \bar{S_i}.\bar{S_j} \delta ^3(r) + \frac{1}{r^3}\left( \frac{3 (\bar{S_i}.{\bar{r}})(\bar{S_j}.{\bar{r}})}{r^2} -\bar{S_i}.\bar{S_j} \right) \right] \left( \bar{ F_i}.\bar{F_j} \right) \nonumber \\ \end{aligned}$$

(64)

$$\begin{aligned} H^{S.O.}(r)= & {} H^{S.O.(c.m.)}(r) + H^{S.O.(t.p)}(r) \end{aligned}$$

(65)

$$\begin{aligned} H^{S.O.(c.m.)}(r)= & {} \frac{\alpha _s(r)}{r^3} \left( \frac{1}{m_i}+\frac{1}{m_j}\right) \left( \frac{S_i}{m_i}+\frac{S_j}{m_j}\right) L \left( \bar{ F_i}.\bar{F_j} \right) \end{aligned}$$

(66)

$$\begin{aligned} H^{S.O.(t.p)}(r)= & {} -\frac{1}{2r}\frac{\delta H^{Conf}}{\delta r}\left( \frac{S_i}{m_i^{2}} + \frac{S_j}{m_j^{2}}\right) L \end{aligned}$$

(67)

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 $\left( l=0\right)$, only the contact term proportional to $\delta ^3(r)$ contributes and the Hamiltonian takes the simpler form as:

$$\begin{aligned} H = \frac{4\alpha _s}{3} \left( -\frac{1}{r} - \frac{8\pi }{3} \delta ^3(r) \frac{\bar{S_i}.\bar{S_j}}{m_im_j}\right) + br + c \end{aligned}$$

(68)

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],

$$\begin{aligned} \langle H_{ss}\rangle = - \frac{8 \pi \alpha _s}{3 m_i m_j} \psi (0)\vert ^2 \equiv - \frac{8 \pi \alpha _s}{3 m_1 m_2} \vert \psi (0)\vert ^2 \end{aligned}$$

(69)

The mass and the decay constant of heavy-flavour pseudo-scalar meson including only spin–spin interaction are given in [26, 27] as:

$$\begin{aligned} M_P = m_1 + m_2 + E_{n,l} + \langle H_{ss}\rangle \end{aligned}$$

(70)

where, $m_1$ and m$_2$ are the masses of quark–antiquark and $\alpha _s$ is the strong coupling constant identified as $\alpha _V$ 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,

$$\begin{aligned} f_P = \sqrt{\frac{12 \vert \psi (0)\vert ^2}{M_P}} \end{aligned}$$

(71)

Here, $\alpha _V$ is the improved strong coupling constant in V-scheme defined by Eq. (4) and $\vert \psi (0)\vert$ is the wave function at origin (WFO).

The non-relativistic binding energy is given by [29, 30, 32],

$$\begin{aligned} E_{n,l}=\frac{-2\mu A^2}{(2(n-1)+1)^2+(2l+1)^2+(4(n-1)+2)(2l+1)}\nonumber \\ \end{aligned}$$

(72)

S-waves states: For the S-wave state $l=0$ , we have;

$$\begin{aligned} E_{n,l}=\frac{-2\mu A^2}{(2(n-1)+1)^2+(4(n-1)+2)+1} \end{aligned}$$

(73)

Now, for 2S state ($n=2$) and for 3S state ($n=3$), the $E_{n,l}$ takes the form as:

$$\begin{aligned} E_{2,0}=\frac{-2\mu A^2}{16} \end{aligned}$$

(74)

and

$$\begin{aligned} E_{3,0}=\frac{-2\mu A^2}{36} \end{aligned}$$

(75)

3 Results

3.1 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. (70) and (71) in 2S and 3S states, which are shown in Tables 6, 7, 8 and 9 respectively. The input parameters are taken as in [24, 36, 37, 52]; $m_{u/d} = 0.336$ GeV, $m _b = 4.95$ GeV, $m_c = 1.55$ GeV, $m_s = 0.483$ GeV, and $b = 0.183$ GeV$^{2}$. Also we calculated the effective strong coupling constant in V-scheme and found $\alpha _{V}({\frac{1}{r}}) =0.73$ for $n_f=4$ and $\alpha _{V}({\frac{1}{r}}) = 0.303$ for $n_f=4$ at $N^3LO$ level respectively. 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 ${\overline{MS}}$ scheme for lowest order (LO level) is given by [40]

$$\begin{aligned} \alpha _s(q^2)=\alpha _{{\overline{MS}}}(q^2)=\frac{4\pi }{\beta _0 ln \left( \frac{m^{2}_Q}{\Lambda _{QCD}^{2}}\right) } \end{aligned}$$

(76)

Here, $m_Q$ is the mass of the heavy quark and $\Lambda _{QCD}$ is the QCD scale parameter having values 0.292 GeV and 0.210 GeV for $n_f=4$ and $n_f=5$ respectively.

3.2 $\alpha _{V}$, $E_{n,l}$ and different parameters used in V-scheme

3.2.1 Calculation of effective strong coupling constant $\alpha _V(\frac{1}{r})$

Using Eq. (4), we tabulate the values of $\alpha _V(\frac{1}{r})$ taking into account one-loop (NLO), two-loop $(N^2LO)$ and three-loop $(N^3LO)$ in 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 1 Values of $\alpha _V(\frac{1}{r})$

3.2.2 Calculation of non-relativistic binding energy $E_{n,l}$

Following Eqs. (74) and (75) along with Eq. (20) we obtain the non-relativistic binding energy $E_{n,l}$ (in GeV) of the heavy-flavour pseudo-scalar mesons D, $D_s$ , B, $B_s,$ $B_c$, $\eta _c$ and $\eta _b$ respectively and tabulated them in Table 2 below.

Table 2 Values $E_{n,l}$ in GeV for 2S and 3S states

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.

3.2.3 Calculation of parameter A

Using Eq. (20) with the values of $\alpha _V$ from Table 1, we calculate the parameter A for the same heavy-flavour pseudo-scalar mesons given in the Table 3 below.

Table 3 Values of parameter A at LO, NLO, $N^2LO$ and $N^3LO$ level

3.2.4 Calculation of different parameters used in the expression of mass $M_P$

For the mentioned seven heavy-flavour pseudo-scalar mesons D, $D_s$ , B, $B_s,$ $B_c$, $\eta _c$ and $\eta _b$ respectively, the Eqs. (18), (19), (20), (32), (33), (34) and (35) along with Eqs. (58), (69) and (70) yield the values of the following parameters (Table 4):

Table 4 Values of parameters used and $\vert \psi (0)\vert ^2$ (in GeV$^3$) in 2S State

Similarly, for the same heavy-flavour pseudo-scalar mesons, the corresponding Eqs. (19), (20), (52), (53), (54), (55), (56), (57) along with Eqs. (59), (69) and (70) yield the following Table 5.

Table 5 Values of parameters used and $\vert \psi (0)\vert ^2$ (in GeV$^3$) in 3S state

3.3 2S State masses and decay constants of heavy-flavour mesons and their comparison

Following Eqs. (58), (70), (71) and (76) along with the results of Tables 1, 2, 3 and 4; we calculate the masses $M_P$ and decay constants $f_P$ (in GeV) of the heavy-flavour pseudo-scalar mesons D, $D_s$ , B, $B_s,$ $B_c$, $\eta _c$ and $\eta _b$ and tabulated them in 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].

Table 6 Mass ($M_P$) in GeV

Table 7 Decay constant ($f_P$) in GeV

3.4 3S State masses and decay constants of heavy-flavour mesons and their comparison

Using Eqs. (59), (70), (71) and (76) along with the results of Tables 1, 2, 3 and 5; we calculate the masses $M_P$ and decay constants $f_P$ (in GeV) of the same heavy-flavour pseudo-scalar mesons and tabulated them in Tables 8 and 9.

Table 8 Mass ($M_P$) in GeV

Table 9 Decay constant ($f_P$) in GeV

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.

3.5 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:

• (a) Due to the numerical values of quark masses. But for heavy-light meson, the uncertainty in the light quark mass does not play any significant role for $m_Q \gg m_q$. i.e., when the heavy quark mass is very large compared to the light quark mass.

• (b) Another uncertainty is in the definition of strong coupling constant in the ${\overline{MS}}$-scheme for lowest order (LO level) defined by Eq. (76). Here the possible uncertainty is in preferring the choice of scale $Q^{2} = m_Q^{2}$ , where $m_Q^{2}$ is the square of the mass of heavy quark. However, even in an alternative choice of the scale $Q^{2} = m_{meson}^{2}$, the differences in the predictions are still insignificant for heavy-light mesons as $m_Q \gg m_q$. For heavy-heavy mesons like $\eta _{c}$ and $\eta _{b}$, where both the quarks are heavy, there will be a very minor change in the predictions of mass spectra and decay constants.

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 $H = M + \frac{p^2}{2m} + \sqrt{p^2+m^2} + V (r)$, 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 ${\overline{C}}^2(\alpha _{V})$ [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].

4 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. The study of mass spectroscopy and decay properties of the heavy-flavour mesonic states considering only spin–spin interaction provides us a fruitful results of the dynamics of quarks and gluons at the hadronic scale. The values of masses are found to be 1.80, 1.886, 5.277, 5.420, 6.45, 2.91 and 9.80 GeV and decay constants are found to be 0.209, 0.238, 0.100, 0.107, 0.311, 0.832, and 0.699 GeV for D, $D_s$, B, $B_s$, $B_c$, $\eta _c$ and $\eta _b$ respectively for pseudo-scalar mesons in 2S state. The corresponding masses and decay constants of the above seven pseudo-scalar mesons for 3S state are found to be 1.860, 1.997, 5.279, 5.425, 6.48, 3.08, 9.87 GeV and 0.100, 0.110, 0.085, 0.087, 0.09, 0.205, 0.200 GeV respectively. The values of $M_P$ and $f_P$ are in close agreement with the values available from other models and experimental data for 2S state. For 3S state the $M_P$ values are in close proximity with other models but some of $f_P$ values are smaller than the other models and experimental data. The success suggests the importance of the effects of three-loop contribution and the non-relativistic binding energy between the quark composite and the choice of renormalization scale parameter used as well as the potential forms for the understanding of the dynamics of the light-heavy and heavy-heavy quark systems. However, although mass spectroscopic results are quite good for 3S state, the values of decay constants $f_P$ are smaller due to the small WFO values. It suggests limitation of the van Royen–Weisskopf formula $(Eq^n.71)$. Proper incorporation of QCD corrections ${\overline{C}}^2(\alpha _{V})$ [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 heavy-flavour mesons with a Coulombic plus linear potential. Moreover, 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.