Revisit the Heavy Vector Quarkonium Leptonic Widths

  • In this paper, we revisit the heavy quarkonium leptonic decays $ \psi(nS) \to \ell^+\ell^- $ and $ \Upsilon(nS) \to \ell^+\ell^- $ using the Bethe-Salpeter method. The emphasis is paid to the relativistic correction. For the cases of $ \psi(1S-5S) $ decay, the relativistic effects are $ 22^{+3}_{-2} $%, $ 34^{+5}_{-5} $%, $ 41^{+6}_{-6} $%, $ 52^{+11}_{-13} $% and $ 62^{+14}_{-12} $%, respectively. And for the cases of $ \Upsilon(1S-5S) $ decay, the relativistic effects are $ 14^{+1}_{-2} $%, $ 23^{+0}_{-3} $%, $ 20^{+8}_{-2} $%, $ 21^{+6}_{-7} $% and $ 28^{+2}_{-7} $%, respectively. Thus the relativistic corrections are large and important for heavy quarkonium leptonic decays, especially for the highly excited charmoniums. Our results of $ \Upsilon(nS) \to \ell^+\ell^- $ consist well with experimental data.
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Guo-Li Wang and Xing-Gang Wu. Revisit the Heavy Vector Quarkonium Leptonic Widths[J]. Chinese Physics C. doi: 10.1088/1674-1137/44/6/063104
Guo-Li Wang and Xing-Gang Wu. Revisit the Heavy Vector Quarkonium Leptonic Widths[J]. Chinese Physics C.  doi: 10.1088/1674-1137/44/6/063104 shu
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Revisit the Heavy Vector Quarkonium Leptonic Widths

    Corresponding author: Guo-Li Wang, gl_wang@hit.edu.cn
    Corresponding author: Xing-Gang Wu, wuxg@cqu.edu.cn
  • 1. Department of Physics, Hebei University, Baoding 071002, China
  • 2. Department of Physics, Chongqing University, Chongqing 401331, China

Abstract: In this paper, we revisit the heavy quarkonium leptonic decays $ \psi(nS) \to \ell^+\ell^- $ and $ \Upsilon(nS) \to \ell^+\ell^- $ using the Bethe-Salpeter method. The emphasis is paid to the relativistic correction. For the cases of $ \psi(1S-5S) $ decay, the relativistic effects are $ 22^{+3}_{-2} $%, $ 34^{+5}_{-5} $%, $ 41^{+6}_{-6} $%, $ 52^{+11}_{-13} $% and $ 62^{+14}_{-12} $%, respectively. And for the cases of $ \Upsilon(1S-5S) $ decay, the relativistic effects are $ 14^{+1}_{-2} $%, $ 23^{+0}_{-3} $%, $ 20^{+8}_{-2} $%, $ 21^{+6}_{-7} $% and $ 28^{+2}_{-7} $%, respectively. Thus the relativistic corrections are large and important for heavy quarkonium leptonic decays, especially for the highly excited charmoniums. Our results of $ \Upsilon(nS) \to \ell^+\ell^- $ consist well with experimental data.

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    1.   Introduction
    • Since it has a clean experimental signal, the dilepton annihilation decay of the heavy vector quarkonium plays an important role in determining the fundamental parameters such as the strong coupling constant [1, 2], the heavy quark masses [1, 35], the heavy quarkonium decay constants [2, 68], and etc. Its decay amplitude is a function of the quarkonium wave function, this process can be used to test various theories such as the quark potential model, the non-relativistic Quantum Chromodynamics (NRQCD), the QCD sums rules, the Lattice QCD, and etc. At present, the Standard Model prediction of the university of lepton flavor is questioned by the measured values of the ratios $ R(D^{(*)}) $ and $ R(K^{(*)}) $ [914], and the quarkonium leptonic decay provides another choice to test the lepton flavor universality.

      The vector quarkonium leptonic decays have been studied for a long time [1521]. Along with the progresses in the computer science and experimental technology, many progresses have been achieved in the literature. For examples, there are Lattice QCD predictions on the leptonic decays of the ground-state Υ and its first radial excitation $ \Upsilon' $ [22]; Ref. [23] gives the next-to-leading nonperturbative prediction and Ref. [24] gives the next-to-leading-log perturbative QCD (pQCD) prediction; Ref. [25] computes the two-loop QCD correction; Ref. [26] studies the inclusive leptonic decay of Υ up to the next-to-next-to-leading order (NNLO) level by including the resummation of logarithms up to (partly) next-to-next-to-leading logarithmic (NNLL) accuracy; and the NNNLO corrections have been discussed by various groups [2731]. A pQCD analysis for $ \Upsilon(1S) $ leptonic decay up to NNNLO level by using the principle of maximum conformality (PMC) [3235] has been done in Refs.[36, 37], in which the renormalization scale ambiguity for the decay width has been eliminated with the help of the renormalization group equation.

      Even though large improvements have been made, there are still deviations between theoretical predictions and experimental data on the heavy vector quarkonium leptonic decays. There are two sources which may cause such deviations. The first one is the unknown higher order perturbative QCD corrections. By using the PMC, the conventional pQCD convergence of the series can be greatly improved due to the elimination of divergent renormalon terms and a more accurate decay width can be achieved, there are still large errors due to unknown high-order terms [36, 37]. The second one is the relativistic correction, which could be large. However almost all the existing pQCD predictions are calculated by using the NRQCD, in which the decay constant of quarkonium, or its wave function at the origin is treated simply within the non-relativistic approximation.

      One may argue that the relativistic correction is small for heavy quarkonium, since the relative velocity among the heavy constitute quarks is small, e.g. $ {v}^2 \sim 0.1-0.3 $ [38]. However, Many analyses done in the literature have found that the relativistic effect could be large. For examples, Bodwin et al. computed the coefficients of the decay operators for the decay of $ ^3 S_1 $ heavy quarkonium into a leptonic pair and found large relativistic correction [39]; Gonzalez et al. pointed out that large relativistic and QCD corrections to the quarkonium leptonic decays are necessary to fit the experimental data [7]; Geng et al. studied the $ B_c $ meson semileptonic decays to charmonium and found that the relativistic corrections are also large [40], especially for highly excited charmonium states. Moreover, from the experimental standpoint, Ref. [41] showed that careful study of leptonic decays is still needed for highly excited charmonium states.

      In this paper, we shall focus on the leptonic decays of charmonium and bottomonium, including their excited states, by using a relativistic method. In a previous short Letter [42], we have finished a relativistic calculation of the quarkonium decays to $ e^{+}e^{-} $, where the results disagree with the experimental data. As a step forward, we shall revisit this topic with more details, supplement the decays to $ \mu^{+}\mu^{-} $ and $ \tau^{+}\tau^{-} $ as well as the ratios of $ R_{\tau\tau} $. We shall show the relativistic effects in these quarkonium decays, and discuss the universality of lepton flavor.

      This remaining parts of the paper are organized as follows. Section II gives the general equation of quarkonium leptonic decay width. In section III, we give a brief review of the Bethe-Salpeter equation, and its instantaneous version, Salpeter equation. We then show the details of how to solve the full Salpeter equation and obtain the relativistic wave function for vector meson in section IV. The calculation of the decay constant in the relativistic method is given in section V. Finally, in section VI, we give the numerical results and discussions. Section VII is reserved for a summary.

    2.   The quarkonium leptonic decay width
    • The leptonic partial decay rate of a vector charmonium or bottomonium $ nS $-state V is given by

      $ \Gamma_{V\to \ell^{+}\ell^{-}} = \frac{4\pi\alpha^2_{em}e_Q^2F^2_V}{3M_{nS}}\times \left(1+2\frac{m^2_{\ell}}{M^2_{nS}}\right)\sqrt{1-4\frac{m^2_{\ell}}{M^2_{nS}}}, $

      (1)

      where $ \alpha_{em} $ is the fine structure constant, $ e_Q $ is the electric charge of the heavy quark Q in unit of the electron charge, $ e_Q = +2/3 $ for charm quark and $ e_Q = -1/3 $ for bottom quark, $ M_{nS} $ is the mass of the $ nS $-state quarkonium, $ m_{\ell} $ is the lepton mass, $ F_V $ is the decay constant of this vector meson that can be defined through the following matrix element of the electromagnetic current

      $ <0|\bar Q\gamma_{\mu} Q|V(P,\epsilon)> = F_V M_{nS}\epsilon_{\mu}, $

      (2)

      where P is quarkonium momentum, and ϵ is the polarization vector.

      In a non-relativistic method, the well-known formula for the decay constant is

      $ F^{NR}_V = \sqrt{\frac{12}{M_{nS}}}|\Psi_V(0)|, $

      (3)

      where $ NR $ means the non-relativistic (NR) result, $ \Psi_V(0) $ is the non-relativistic wave function evaluated at the origin. In this NR method, only one single radial wave function exists, and this vector meson and its corresponding pseudoscalar one have the same radial wave function and same decay constant. But in a relativistic method, they will have different wave functions and different decay constants, we will see that more than one radial wave functions have contributions to the vector meson decay constant.

      In a relativistic method, the decay constant $ F_V = F_V^{Re} $ is not relate to the wave function at origin, but the full region. In the following part, we will focus on the calculating of the $ F_V^{Re} $ by a relativistic method.

    3.   Bethe-salpeter equation and salpeter equation
    • In this section, we briefly review the Bethe-Salpeter (BS) equation [43], which is a relativistic dynamic equation describing the two-body bound state, and its instantaneous version, Salpeter equation [44]. The BS equation for a meson, which is a bound state of a quark labelled as 1 and anti-quark labelled as 2, can be written as [43]

      $ ({\not\!\! p}_{1}-m_{1})\chi_{P}(q)({\not \!\!p}_{2}+m_{2}) = i\int\frac{d^{4}k}{(2\pi)^{4}}V(P,k,q)\chi_{P}(k)\;, $

      (4)

      where $ \chi_{P}(q) $ is the relativistic wave function of the meson, $ V(P,k,q) $ is the interaction kernel between the quark and anti-quark, $ p_{1}, p_{2},m_1,m_2 $ are the momenta and masses of the quark and anti-quark, P is the momentum of the meson, q is the relative momentum between quark and anti-quark. The momenta $ p_1 $ and $ p_2 $ satisfy the relations, $ p_{1} = {\alpha}_{1}P+q $ and $ p_{2} = {\alpha}_{2}P-q $, where $ {\alpha}_{1} = \frac{m_{1}}{m_{1}+m_{2}} $ and $ {\alpha}_{2} = \frac{m_{2}}{m_{1}+m_{2}} $. For the case of quarkonium, in which $ m_1 = m_2 $, we have $ {\alpha}_{1} = {\alpha}_{2} = 0.5 $.

      In general cases, the BS equation is hard to be solved due to the complex interaction kernel between the constituent quarks. For the present considered doubly heavy quarkonium, the interaction kernel between two heavy constituent quarks can be treated as an instantaneous one, leading to a simpler instantaneous version of the BS equation. For this case, it is convenient to divide the relative momentum q into two parts, $ q^{\mu} = q^{\mu}_{\parallel}+q^{\mu}_{\perp} $, where $ q^{\mu}_{\parallel}\equiv (P\cdot q/M^{2})P^{\mu} $ and $ q^{\mu}_{\perp}\equiv q^{\mu}-q^{\mu}_{\parallel} $, M is the mass of the bound state, and we have $ P^2 = M^2 $. Then we have two Lorentz invariant variables, $ q_{P} = \frac{(P\cdot q)}{M} $ and $ q_{T} = \sqrt{q^{2}_{P}-q^{2}} = $$ \sqrt{-q^{2}_{\perp}} $, when $ \vec{P} = 0 $, that is in the center-of-mass system of the meson, they turn to the usual component $ q_{0} $ and $ |\vec q| $, and $ q_{\perp} = (0,\vec{q}) $.

      With these notations, the volume element of the relativistic momentum k can be written in an invariant form $ d^4k = dk_{P} k^2_{T} {{dk_{T}}} ds d\phi $, where $ ds = (k_{P} q_{P} -k\cdot q)/(k_{T} q_{T}) $ and $ \phi $ is the azimuthal angle. Taking the instantaneous approximation, in the center of mass frame of the bound state, the kernel $ V(P,k,q) $ changes to $ V(k_{\perp},q_{\perp},s) $, then we introduce the three-dimensional wave function

      $ \Psi_{P}(q^{\mu}_{\perp})\equiv i\int \frac{dq_{P}}{2\pi}\chi(q^{\mu}_{\parallel},q^{\mu}_{\perp}), $

      (5)

      and the notation

      $ \eta(q^{\mu}_{\perp})\equiv\int\frac{k^2_{T}dk_{T} ds {{d \phi}}}{(2\pi)^{{{3}}}} V(k_{\perp},q_{\perp},s)\Psi_{P}(k^{\mu}_{\perp}). $

      (6)

      Then the BS Eq. (4) is rewritten as

      $ \chi(q_{\parallel},q_{\perp}) = S_{1}(p_{1})\eta(q_{\perp})S_{2}(p_{2}), $

      (7)

      where $ S_{1}(p_{1}) $ and $ S_{2}(p_{2}) $ are propagators of quark 1 and anti-quark 2, respectively, which can be decomposed as

      $ S_{i}(p_{i}) = \frac{\Lambda^{+}_{i}(q_{\perp})}{(-1)^{i+1} q_{P} +\alpha_{i}M-\omega_{i}+i\varepsilon}+ \frac{\Lambda^{-}_{i}(q_{\perp})}{(-1)^{i+1} q_{P}+\alpha_{i}M+\omega_{i}-i\varepsilon}\;. $

      (8)

      Here we have defined the constituent quark energy $ \omega_{i} = \sqrt{m_{i}^{2}+q^{2}_{T}} $ and the project operator $ \Lambda^{\pm}_{i}(q_{\perp}) = $$ \frac{1}{2\omega_{i}}\left[ \frac{{\not P}}{M}\omega_{i}\pm (-1)^{i+1} (m_{i}+{\not q}_{\perp})\right] $, where $ i = 1 $ and 2 for quark and anti-quark, respectively.

      Using the project operators, we can divide the wave function into four parts

      $ \Psi_{P}(q_{\perp}) = \Psi^{++}_{P}(q_{\perp})+ \Psi^{+-}_{P}(q_{\perp})+\Psi^{-+}_{P}(q_{\perp}) +\Psi^{--}_{P}(q_{\perp}), $

      (9)

      with the definition $ \Psi^{\pm\pm}_{P}(q_{\perp})\equiv \Lambda^{\pm}_{1}(q_{\perp}) \frac{{\not P}}{M}\Psi_{P}(q_{\perp}) $$ \frac{{\not P}}{M} \Lambda^{{\pm}}_{2}(q_{\perp}) $. Here $ \Psi^{++}_{P}(q_{\perp}) $ and $ \Psi^{--}_{P}(q_{\perp}) $ are usually called as the positive and negative energy wave functions of the quarkonium.

      After integrating over $ q_{P} $ on both sides of Eq. (7) with contour integrations, we obtain the famous Salpeter equation [44]:

      $ \Psi_{P}(q_{\perp}) = \frac{ \Lambda^{+}_{1}(q_{\perp})\eta(q_{\perp})\Lambda^{+}_{2}(q_{\perp})} {(M-\omega_{1}-\omega_{2})}- \frac{ \Lambda^{-}_{1}(q_{\perp})\eta(q_{\perp})\Lambda^{-}_{2}(q_{\perp})} {(M+\omega_{1}+\omega_{2})}. $

      (10)

      Or equally, the Salpeter equation can be written as four independent equations by using the project operators:

      $ (M-\omega_{1}-\omega_{2})\Psi^{++}_{P}(q_{\perp}) = \Lambda^{+}_{1}(q_{\perp})\eta(q_{\perp})\Lambda^{+}_{2}(q_{\perp})\;, $

      (11)

      $ (M+\omega_{1}+\omega_{2})\Psi^{--}_{P}(q_{\perp}) = - \Lambda^{-}_{1}(q_{\perp})\eta(q_{\perp})\Lambda^{-}_{2}(q_{\perp})\;, $

      (12)

      $ \Psi^{+-}_{P}(q_{\perp}) = 0\;, $

      (13)

      $ \Psi^{-+}_{P}(q_{\perp}) = 0\;. $

      (14)

      The normalization condition for BS wave function is read as

      $ \int\frac{q_{T}^2dq_{T}}{2{\pi}^2}Tr\left[\overline\Psi^{++}_{P} \frac{{/}\!\!\! {P}}{M}\Psi^{++}_{P}\frac{{/}\!\!\!{P}}{M}-\overline\Psi^{--}_{P} \frac{{/}\!\!\! {P}}{M}\Psi^{--}_{P}\frac{{/}\!\!\! {P}}{M}\right] = 2M\;. $

      (15)

      Note that in the literature, usually not the full Salpeter Eq. (10) (or equally, the four Eqs. (11-14)), but the first Eq. (11) is solved, where only the positive wave function is involved. There is a logical reason to make such approximation, i.e. in its effective range, the numerical value of $ M-\omega_{1}-\omega_{2} $ in Eq. (11) is much smaller than that of the corresponding $ M+\omega_{1}+\omega_{2} $ in Eq. (12), thus results in dominant value of positive wave function $ \Psi^{++}_P(q_{\perp}) $, and then the contribution of negative wave function $ \Psi^{--}_P(q_{\perp}) $ can be safely neglected. However we should point out that if we only consider Eq. (11) of $ \Psi^{++}_P(q_{\perp}) $, we ignore not only the contribution of the negative wave function, but also the relativistic effects of these wave functions. The reason lies in that the number of eigenvalue equations will limit the number of the radial wave functions, and as shall be shown below, only the coupled four Eqs. (11-14) can provide us enough information to derive a relativistic wave function.

    4.   Relativistic wave function and the kernel
    • Though the BS or the Salpeter equation is the relativistic dynamic equation describing the two-body bound state, the equation itself cannot provide us the information of the wave function automatically. This means that we have to provide an explicit form for the relativistic kinematic wave function as an input, which could be constructed by using all the allowable Lorentz- and γ- structures.

      In the literature, we are familiar with the form of the non-relativistic wave function for a $ 1^- $ vector, e.g.

      $ \Psi_P(\vec{q}) = ({\not \!\!P}+M){\not {\epsilon}} \psi(\vec{q}), $

      (16)

      where M, P and ϵ are mass, momentum and polarization of the vector meson, respectively, $ \vec{q} $ is the relative momentum between the quark and anti-quark. There is only one unknown wave function $ \psi(\vec{q}) $ in Eq.(16), which can be obtained numerically by solving the BS Eq. (11) or the non-relativistic Schrödinger equation. The relative momentum $ \vec{q} $ is related to the relative velocity $ \vec v $ between quark and anti-quark in the meson, $ \vec q = \frac{m_1 m_2}{m_1+m_2}\vec {v} $. A relativistic wave function should depend on the relative velocity $ \vec {v} $ or momentum $ \vec{q} $ separately, not merely on the radial part $ \psi(\vec {q}) $, because the radial part actually is $ \psi(|\vec {q}|) $ or equally $ \psi(\vec {q}^2) $.

      To give the relativistic wave function form, we start from the $ J^{pc} $ of a meson, because the $ J^{p} $ or $ J^{pc} $ is a good quantum number in any case, where J is total angular momentum, p and c are parity and charge conjugate parity of the meson. Parity transform changes the momentum $ q = (q_0,\vec{q}) $ to $ q' = (q_0,-\vec{q}) $, so for a meson, after applying the parity transform, its four-dimensional wave function $ \chi_{P}(q) $ changes to $ p\cdot\gamma_0\chi_{{P'}}(q')\gamma_0 $, where p is the eigenvalue of parity. Charge conjugate transform changes $ \chi_{P}(q) $ to $ c\cdot {\cal{C}}\chi^{T}_{P}(-q){\cal{C}}^{-1} $, where c is the eigenvalue of charge conjugate parity, and $ {\cal{C}} = \gamma_2\gamma_0 $ is the charge conjugate transform operator, T is the transpose transform. Since the Salpeter equation is instantaneous, the input wave function $ \Psi_{P}(q_{\perp}) $ is also instantaneous, and the general form of the wave function for a $ 1^{-} $ vector meson can be written as [45, 46]

      $ \begin{split} \Psi_{P}^{1^{-}}(q_{\perp}) =& q_{\perp}\cdot{\epsilon}_{\perp} \left[\psi_1(q_{\perp})+\frac{\not\!\!P}{M}\psi_2(q_{\perp})+ \frac{{\not\!\!q}_{\perp}}{M}\psi_3(q_{\perp})\right.\\&\left.+\frac{{\not\!\!P} {\not\!q}_{\perp}}{M^2} \psi_4(q_{\perp})\right]+ M{\not\!\epsilon}_{\perp}\psi_5(q_{\perp}) \\ &+ {\not\!\epsilon}_{\perp}{\not\!P}\psi_6(q_{\perp})+ ({\not\!q}_{\perp}{\not\!\epsilon}_{\perp}- q_{\perp}\cdot{\epsilon}_{\perp}) \psi_7(q_{\perp})\\&+\frac{1}{M}({\not\!\!P}{\not\!\epsilon}_{\perp} {\not\!\!q}_{\perp}-{\not\!P}q_{\perp}\cdot{\epsilon}_{\perp}) \psi_8(q_{\perp}). \end{split} $

      (17)

      There are totally 8 radial wave functions $ \psi_i(q_{\perp}) = \psi_i(|\vec q|) $ with $ i = 1\sim8 $, which obviously can not be obtained by solving only one equation, e.g. Eq. (11), but can be obtained by solving the full Salpeter Eqs. (11-14). The above expression does not include the terms with $ P\cdot q $, since in the condition of instantaneous interaction, $ P\cdot q = P \cdot q_{\perp} = 0 $. There are also no higher order $ q_{\perp} $ terms like $ q^2_{\perp} $, $ q^3_{\perp} $, $ q^4_{\perp} $, etc, because the even power of $ q_{\perp} $ can be absorbed into the radial part of $ \psi_i(q_{\perp}) $, while the odd power $ q_{\perp} $ terms can be changed to a lower power one, for example, $ {\not\!q}^3_{\perp}\psi'_i(q_{\perp}) = {\not\!q}_{\perp}\psi_i(q_{\perp}) $. By the way, if we delete all the $ q_\perp $ terms except those inside the radial wave functions, then the wave function Eq. (17) reduces to $ M{\not\!\epsilon}_{\perp}\psi_5(q_{\perp})+{\not\!\epsilon}_{\perp}{\not\!P}\psi_6(q_{\perp}) $, and by further setting $ \psi_5(q_{\perp}) = -\psi_6(q_{\perp}) = \psi(q_{\perp}) $, the wave function reduces to the non-relativistic one, e.g. Eq. (16). Thus the terms of $ \psi_1 $, $ \psi_2 $, $ \psi_3 $, $ \psi_4 $, $ \psi_7 $ and $ \psi_8 $ in Eq. (17) are all relativistic corrections.

      When we consider the charge conjugate parity, the terms with $ \psi_2(q_{\perp}) $ and $ \psi_7(q_{\perp}) $ vanish because of positive charge conjugate parity $ c = + $, and the general instantaneous wave function for a $ 1^{--} $ quarkonium becomes

      $ \begin{split} \Psi_{P}^{1^{--}}(q_{\perp}) =& q_{\perp}\cdot{\epsilon}_{\perp} \left[\psi_1(q_{\perp})+ \frac{{\not\!q}_{\perp}}{M}\psi_3(q_{\perp})+\frac{{\not\!P} {\not\!q}_{\perp}}{M^2} \psi_4(q_{\perp})\right]\\&+ M{\not\!\epsilon}_{\perp}\psi_5(q_{\perp}) + {\not\!\epsilon}_{\perp}{\not\!P}\psi_6(q_{\perp})\\ &+\frac{1}{M}({\not\!P}{\not\!\epsilon}_{\perp} {\not\!q}_{\perp}-{\not\!P}q_{\perp}\cdot{\epsilon}_{\perp}) \psi_8(q_{\perp}). \end{split} $

      (18)

      Before going on, we would like to discuss the interaction kernel $ {{V(r)}} $. We know that from Quantum Chromodynamics, the strong interaction between quark and antiquark is given by exchange of gluon(s), and the basic kernel contains a short-range $ \gamma_{\mu}\otimes\gamma^{\mu} $ vector interaction $ -\frac{4\alpha_s}{3r} $ plus a long-range $ 1\otimes 1 $ linear confining scalar interaction $ \lambda r $ suggested by Lattice QCD calculations [47]. In Coulomb gauge and leading order, the kernel is the famous Cornell potential

      $ V(r) = \lambda r+V_0-\gamma_0\otimes\gamma^0\frac{4}{3}\frac{\alpha_s}{r}, $

      (19)

      where λ is the string tension, $ V_0 $ is a free constant appearing in potential model to fit data, $ \alpha_s $ is the running coupling constant. In order to avoid infrared divergence and incorporate the screening effects, one can add an exponential factor $ e^{-\alpha r} $ to the potential [48], i.e.

      $ V(r) = \frac{\lambda}{\alpha}(1-e^{-\alpha r})+V_0-\gamma_0\otimes\gamma^0\frac{4}{3}\frac{\alpha_s}{r}e^{-\alpha r}. $

      (20)

      It is easy to check that when $ \alpha r\ll 1 $, Eq. (20) returns to Eq. (19). In the momentum space and the rest frame of the bound state, the potential takes the form:

      $ V(\vec{q}) = V_s(\vec{q})+\gamma_{0}\otimes\gamma^0 V_v(\vec{q}), $

      (21)

      where

      $ \begin{split} V_s(\vec{q}) =& -(\frac{\lambda}{\alpha}+V_0) \delta^3(\vec{q})+\frac{\lambda}{\pi^2} \frac{1}{(\vec{q}^2+{\alpha}^2)^2},\\ V_v(\vec{q}) =& -\frac{2}{3{\pi}^2}\frac{\alpha_s( \vec{q})}{{(\vec{q}}^2+{\alpha}^2)}, \\ \alpha_s(\vec{q}) =& \frac{12\pi}{33-2N_f}\frac{1} {\log (e+\frac{{\vec{q}}^2}{\Lambda^{2}_{QCD}})}. \end{split} $

      Here $ \alpha_s(\vec{q}) $ is the running coupling with one loop QCD correction, and $ e = 2.71828 $. The constants λ, α, $ V_0 $ and $ \Lambda_{QCD} $ are the parameters which characterize the potential, and $ N_f = 3 $ for $ c {\bar c} $ system, $ N_f = 4 $ for $ b {\bar b} $ system.

      The readers may have a question why we choose a simple basic kernel, and not a relativistic one [47, 49] which includes details of spin-independent potential and spin-dependent potential like the spin-spin interaction, spin-orbital interaction, tensor interaction, etc. The reason is that in our relativistic method, with a relativistic wave function for the bound state, we only need the basic potential and not a relativistic one, otherwise we will meet the double counting problem. To explain this, let us show how the relativistic potential is obtained: the potential between a quark and an anti-quark is constructed by on-shell $ q \bar q $ scattering amplitude in the center-of-mass frame motivated by single gluon exchange, where the gluon propagator is given in the Coulomb gauge. From the amplitude, at leading-order level, the basic non-relativistic vector potential $ -\frac{4\alpha_s}{3r} $ is obtained (usually in momentum space). To obtain the relativistic corrections of the potential, the on-shell Dirac spinors of the quark and anti-quark are expanded according to quantities like mass, momentum, etc, then the relativistic potential is obtained; at the same time, the relativistic corrections from the free spinors (wave function for a bound state) are moved to the potential, then the corresponding wave function become to a non-relativistic one.

      In our case, we have a relativistic wave function, then the potential is non-relativistic, otherwise both of them are relativistic, there is the double counting. So generally, a relativistic method, can be a relativistic wave function with a non-relativistic potential, or a non-relativistic wave function with a relativistic potential. In principle, a half-relativistic wave function with a half-relativistic potential is also permitted, but one has to be careful to avoid double counting. Usually, the method with a non-relativistic wave function and a relativistic potential, is good at calculating the mass spectrum of bound states; while the method with a relativistic wave function and a non-relativistic potential, not only good at calculating the mass spectrum as a eigenvalue problem, but also good at calculating transition amplitude.

      Now with the kernel Eq. (21) and relativistic wave function Eq. (17) or Eq. (18), we are ready to solve the coupled Salpeter Eqs. (11-14). Substituting the wave function Eq. (18) into Eq. (13) and Eq. (14), taking trace on both sides, multiplying a quantity with polarization vector on both sides, e.g. $ q_{\perp}\cdot \epsilon^{*} $ or $ {{\not {\epsilon}}^{*}_{\perp}}\cdot {\not P} $, and then by using the completeness of the polarization vector, we obtain the relations

      $ \psi_1(q_{\perp}) = \frac{q_{\perp}^2 \psi_3(q_{\perp}) +M^2\psi_5(q_{\perp})} {Mm_1},\,\; \; \; \psi_8(q_{\perp}) = -\frac{\psi_6(q_{\perp})M} {m_1}, $

      where we have used $ m_1 = m_2 $ for quarkonium state. Now we have only four independent unknown radial wave functions, $ \psi_3(q_{\perp}) $, $ \psi_4(q_{\perp}) $, $ \psi_5(q_{\perp}) $, $ \psi_6(q_{\perp}) $, whose numerical values can be obtained by solving Eq. (11) and Eq. (12). Substituting the wave function Eq. (18) into Eq. (11) and Eq. (12), repeating the steps of taking trace, we finally obtain four coupled equations

      $ \begin{split}& (M-2\omega_1)\left\{\left(\psi_3(\vec {q})\frac{\vec {q}^2}{M^2}-\psi_5(\vec {q})\right) +\left(\psi_4(\vec {q})\frac{\vec {q}^2}{M^2}+\psi_6(\vec {q})\right)\frac{m_1} {\omega_1}\right\} = \int{\frac{d^3\vec{k}}{(2\pi)^3}\frac{2}{\omega_1^2}}\left\{(V_s+V_v) \left(\psi_3(\vec{k})\frac{{\vec{k}}^2}{M^2}-\psi_5(\vec{k})\right) (\vec{k}\cdot\vec{q})\right.\\&\quad -(V_s-V_v)\left[m_1^2\left(\psi_3(\vec{k})\frac{(\vec{k}\cdot \vec{q})^2}{M^2\vec {q}^2}-\psi_5(\vec{k})\right)\left.+m_1\omega_1\left(\psi_4(\vec{k})\frac{(\vec{k}\cdot \vec{q})^2}{M^2\vec {q}^2}+\psi_6(\vec{k})\right)\right]\right\}\,, \end{split} $

      (22)

      $ \begin{split}& (M+2\omega_1)\left\{\left(\psi_3(\vec {q})\frac{\vec {q}^2}{M^2}-\psi_5(\vec {q})\right) -\left(\psi_4(\vec {q})\frac{\vec {q}^2}{M^2}+\psi_6(\vec {q})\right)\frac{m_1} {\omega_1}\right\} = -\int{\frac{d^3\vec{k}}{(2\pi)^3}\frac{2}{\omega_1^2}}\left\{(V_s+V_v)\left[ \left(\psi_3(\vec{k})\frac{{\vec{k}}^2}{M^2}-\psi_5(\vec{k})\right) \right](\vec{k}\cdot\vec{q})\right. \\ &\quad -(V_s-V_v)\left[m_1^2\left(\psi_3(\vec{k})\frac{(\vec{k}\cdot \vec{q})^2}{M^2\vec {q}^2}-\psi_5(\vec{k})\right)\left.-m_1\omega_1\left(\psi_4(\vec{k})\frac{(\vec{k}\cdot \vec{q})^2}{M^2\vec {q}^2}+\psi_6(\vec{k})\right)\right]\right\}\,, \end{split} $

      (23)

      $ \begin{split}& (M-2\omega_1)\left\{\left(\psi_3(\vec {q})+\psi_4(\vec {q})\frac{m_1}{\omega_1}\right) \frac{\vec {q}^2}{M^2} -3\left(\psi_5(\vec {q})-\psi_6(\vec {q})\frac {\omega_1}{m_1}\right)-\psi_6(\vec {q})\frac{\vec {q}^2}{m_1\omega_1}\right\} =\\&\quad -\int{\frac{d^3\vec{k}}{(2\pi)^3}\frac{1}{\omega_1^2}}\left\{(V_s+V_v) \left[-\frac{2\omega_1}{m_1} \psi_6(\vec{k})-\psi_3(\vec{k})\frac{{\vec k}^2}{M^2}+ \psi_5(\vec{k})\right](\vec{k}\cdot \vec{q})\right. \\ &\quad +(V_s-V_v)\left[\omega_1^2\left(\psi_3(\vec{k})\frac{{\vec k}^2}{M^2} -3\psi_5(\vec{k})\right)+m_1\omega_1 \left(\psi_4(\vec{k})\frac{{\vec k}^2}{M^2}+3\psi_6(\vec{k})\right)\right. \left.\left. -\left(\psi_3(\vec{k})\frac{(\vec{k}\cdot\vec{q})^2}{M^2} -\psi_5(\vec{k})\vec{q}^2\right)\right]\right\}\,, \end{split} $

      (24)

      $ \begin{split}& (M+2\omega_1)\left\{\left[\psi_3(\vec {q})-\psi_4(\vec {q})\frac{m_1}{\omega_1}\right] \frac{\vec {q}^2}{M^2}-3\left(\psi_5(\vec {q})+\psi_6(\vec {q})\frac {\omega_1}{m_1}\right)+\psi_6(\vec {q})\frac{\vec {q}^2}{m_1\omega_1}\right\} =\\&\quad \int{\frac{d^3\vec{k}}{(2\pi)^3}\frac{1}{\omega_1^2}}\left\{(V_s+V_v) \left[\frac{2\omega_1}{m_1}\psi_6(\vec{k}) -\psi_3(\vec{k})\frac{{\vec k}^2}{M^2}+ \psi_5(\vec{k})\right](\vec{k}\cdot\vec{q})\right. \\ &\quad +(V_s-V_v)\left[\omega_1^2\left(\psi_3(\vec{k})\frac{{\vec k}^2}{M^2}-3\psi_5(\vec{k})\right)-m_1\omega_1 \left(\psi_4(\vec{k})\frac{{\vec k}^2}{M^2}+3\psi_6(\vec{k})\right)\right. \left.\left. -\left(\psi_3(\vec{k})\frac{(\vec{k}\cdot\vec{q})^2}{M^2} -\psi_5(\vec{k}){\vec {q}^2}\right)\right]\right\}\,, \end{split} $

      (25)

      where we have used the relation $ \omega_1 = \omega_2 $ for quarkonium, and $ V_s = V_s(\vec q - \vec k) $, $ V_v = V_v(\vec q - \vec k) $. Since we have four coupled equations, numerical values of the four independent radial wave functions can be obtained, and as an eigenvalue problem, the mass spectrum can be obtained simultaneously.

      Now the normalization condition Eq. (15) for the $ 1^{--} $ wave function is

      $ \begin{split} & \int \frac{d^3{\vec q}}{(2\pi)^3}\frac{8\omega_1}{3M}\left\{ 3\psi_5(\vec {q})\psi_6(\vec {q})\frac{M^2}{2m_1}\right. +\frac{\vec q^2}{2m_1}\\&\quad\left.\left[ \psi_4(\vec {q})\psi_5(\vec {q})-\psi_3(\vec {q})\left(\psi_4(\vec {q})\frac{\vec q^2}{M^2}+\psi_6(\vec {q})\right)\right] \right\} = 1. \end{split}$

      (26)
    5.   The decay constant in salpeter method
    • The relativistic decay constant $ F^{Re}_{V} $ of Eq. (2) for a vector quarkonium in BS method can be calculated as

      $ \begin{split} F^{Re}_V M\epsilon_{\mu} =& \sqrt{N_c}\int\frac{d^4q}{(2\pi)^4}{\rm{Tr}}[\chi_{P}({q})\gamma_\mu] \\=& i\sqrt{N_c}\int\frac{d^3{\vec q}}{(2\pi)^3}{\rm{Tr}}[\Psi_{P}({\vec q})\gamma_\mu], \end{split}$

      (27)

      where $ N_c = 3 $ is the color number, Tr is the trace operator. We note that, in calculating the decay constant, the Salpeter wave function $ \Psi_{P}({\vec q}) $, not merely the positive wave function $ \Psi^{++}_{P}({\vec q}) $ has contribution. For a vector quarkonium with the relativistic wave function Eq. (18), we obtain the relativistic decay constant

      $ \begin{split} F^{Re}_{V} = 4\sqrt{3} \int \frac{d^3 \vec{q}}{(2\pi)^3} \left[\psi_{5}({\vec q})-\frac{{\vec q}^2}{3M^2}\psi_3(\vec{q})\right], \end{split} $

      (28)

      where we also note that, not only the $ \psi_{5} $, but also the $ \psi_{3} $ term has contribution.

    6.   Results and discussions

      6.1.   Input parameters and the heavy quarkonium wave functions

    • The input parameters can be fixed by fitting the mass spectra of charmonium and bottomonium. We choose $ m_b = 4.96\; {\rm{GeV}} $, $ m_c = 1.60\; {\rm{GeV}} $, $ \alpha = 0.06 $ GeV, and $ \Lambda_{\rm QCD} = 0.21 $ GeV. At the same time, we choose $ \lambda = 0.23 \;{\rm{GeV}} ^2 $ and $ V_0 = -0.249 $ GeV for charmonium system and $ \lambda = 0.2 \;{\rm{GeV}}^2 $ and $ V_0 = -0.124 $ GeV for bottomonium system.

      The mass spectra of vector charmonium and bottomonium are shown in Table I. The theoretical predictions are consistent with experimental data given by Particle Data Group (PDG). For the wave functions, as an example, we present four different $ J/\psi $ radial wave functions in Figure 1, where the dominant radial wave functions $ \psi_5 $ and $ \psi_6 $, and two minor ones $ {\vec q}^2\psi_3/M^2 $ and $ {\vec q}^2\psi_4/M^2 $ are given. From now on, we will use the symbols $ |\vec q| = q $ and $ |\vec v| = v $ for simplicity.

      nSTh($ c{\bar c} $)Exp($ c{\bar c} $)Th($ b{\bar b} $)Exp($ b{\bar b} $)
      1S3097.33096.99460.79460.3
      2S3686.43686.110020.510023.3
      3S4059.3403910362.610355.2
      4S4337.5442110622.210579.4
      5S4559.410835.110889.9

      Table 1.  Mass spectra of the S wave $ c{\bar c} $ and $ b{\bar b} $ vectors in unit of MeV. 'Th' is the theoretical prediction, 'Exp' is the experimental data from PDG [50].

      Figure 1.  Diagrams of the four typical radial wave functions of $ J/\psi $.

      As we have described in Sec IV, the terms with radial wave functions $ \psi_5 $ and $ \psi_6 $ in the wave function, e.g. Eq.(18), are non-relativistic ones, and all the others are relativistic corrections. Figure 1 may indicate that the relativistic wave functions, $ \psi_3 $ and $ \psi_4 $, can be safely neglected, but it is not true. Figure 1 only shows the relative importance of the wave functions in small q region, and to see the relative importance of the wave functions in whole q-region, we plot the ratio $ \psi_5/({q}^2\psi_3/M^2) $ in Figure 2. It shows that in large q region, the value of $ \psi_5 $ is only several times larger than that of the $ ({q}^2\psi_3/M^2) $. Thus the terms which are proportional to $ \psi_{3} $ ($ \psi_{4} $ and others) may has sizable contributions in large q region, leading to possibly sizable relativistic corrections.

      Figure 2.  Ratio $ \frac{\psi_5}{{q}^2\psi_3/M^2} $ of the $ J/\psi $ radial wave functions.

    • 6.2.   Charmonium leptonic decay widths

    • Our results of $ \psi(nS)\to\ell^+\ell^- $ are shown in Table II, where in the second column, 'NR', the non-relativistic decay rates are shown, that means in Eq. (28), the $ \psi_3 $ term is ignored, only $ \psi_5 $ has contribution. Third column 'Re' show the relativistic results including the contribution both of $ \psi_5 $ and $ \psi_3 $. One can see that the relativistic results are deferent from the non-relativistic ones obviously in charmonium cases. To see this clearly, we add the fourth column '(NR-Re)/Re' in Table II to give the ratio (NR-Re)/Re, whose value can be called as the 'relativistic effect'.

      modesNRRe$ \frac{\rm{NR-Re}}{\rm{Re}} $Exp
      $ J/\psi\to e^{+}e^{-} $$ 10.95^{+2.20}_{-1.86} $$ 8.95^{+1.57}_{-1.38} $$ 22.3^{+2.7}_{-2.2} $%5.55$ \pm $0.16
      $ \psi(2S)\to e^{+}e^{-} $$ 5.92^{+1.05}_{-0.89} $$ 4.43^{+0.60}_{-0.54} $$ 33.6^{+5.0}_{-4.3} $%$ 2.33\pm $0.04
      $ \psi(2S)\to \tau^{+}\tau^{-} $$ 2.31^{+1.68}_{-2.31} $$ 1.73^{+1.29}_{-1.73} $$ 33.6^{+3.2}_{-4.1} $%0.91$ \pm $0.14
      $ \psi(3S)\to e^{+}e^{-} $$ 4.30^{+0.69}_{-0.66} $$ 3.04^{+0.35}_{-0.35} $$ 41.4^{+5.6}_{-6.1} $%0.86$ \pm $0.07
      $ \psi(3S)\to\tau^{+}\tau^{-} $$ 2.87^{+0.61}_{-1.23} $$ 2.03^{+0.48}_{-0.87} $$ 41.4^{+6.9}_{-6.3} $%
      $ \psi(4S)\to e^{+}e^{-} $$ 3.53^{+0.65}_{-0.66} $$ 2.32^{+0.24}_{-0.26} $$ 52.2^{+11.1}_{-12.9} $%0.48$ \pm $0.22
      $ \psi(4S)\to\tau^{+}\tau^{-} $$ 2.70^{+0.26}_{-0.60} $$ 1.78^{+0.25}_{-0.43} $$ 52.2^{+15.4}_{-13.0} $%
      $ \psi(5S)\to e^{+}e^{-} $$ 3.05^{+0.55}_{-0.52} $$ 1.88^{+0.16}_{-0.19} $$ 62.2^{+14.3}_{-12.5} $%0.58$ \pm $0.07
      $ \psi(5S)\to\tau^{+}\tau^{-} $$ 2.49^{+0.30}_{-0.58} $$ 1.54^{+0.21}_{-0.31} $$ 62.2^{+16.4}_{-23.1} $%

      Table 2.  Decay rates of $ \psi(nS)\to\ell^+\ell^- $ in unit of keV. 'NR' means the non-relativistic result, 'Re' is the relativistic result, 'Exp' is the experimental data from PDG [50].

      Table II indicates that the relativistic effect is about 22% for $ J/\psi $ decay, which is consist with our normal power counting for relativistic terms, e.g. $ v_c^2 \sim 0.2-0.3 $. While for excited states, the relativistic effects are much larger than that of the ground state. For 2S, 3S, 4S and 5S states, the relativistic effects are about 34%, 41%, 52% and 62%, respectively. These results are consistent with previous conclusion when we study the semi-leptonic decays $ B^+_c \to {c{\bar c}}+\ell^+ +\nu_{\ell} $, higher excited charmonium state has larger relativistic effect [40]. This conclusion can also be obtained qualitatively from the diagrams of radial wave functions. We mention that the relative momentum q relates to the relative velocity $ v_Q $ between quark and antiquark in the quarkonium, $ q = 0.5 m_Q v_Q $. As shown by Figure 1, two non-relativistic $ J/\psi $ radial wave functions are always dominant over the relativistic ones in whole q region, leading to a small relativistic correction. While for the excited states, see Figure 3 as an example, where we plot the radial wave functions of $ \psi(2S) $, the non-relativistic ones are still dominant in small q region, but there is a node structure for each curve. Before the node, q is small, after the node, alone with q become large, the wave function will change its sign. Contributions of the wave functions in lower q region before and after the node will cancel each other out, and the wave function in larger q ($ v_Q $) region may give sizable contribution, resulting in large relativistic correction.

      Figure 3.  Diagrams of the radial wave functions of $ \psi(2S) $.

      There are other methods that also consider the relativistic effect in the heavy quarkonium decays. For example, Bodwin et al. [39] and Brambilla et al. [51] computed the $ {v_Q}^2 $- and the $ {v_Q}^4 $- corrections to the decay rate of $ Q{\bar{Q}} $-quarkonium within the framework of NRQCD. For the case of $ J/\psi $ [39], the predicted relativistic effect is $ 34.1 $% for $ v_c^2\sim 0.3 $, which changes to 23.0% for $ {v_c}^2\sim 0.18 $; those values are consistent with our present prediction of 22.3%.

      In Table II, we also show the theoretical uncertainties caused by the choices of input parameters, we vary all the parameters simultaneously within $ \pm 10 $% of their central values, and take the largest variation as the uncertainties. Within the errors, most of predictions are much larger than the experimental data, and the only exception is the channel of $ \psi(2S)\to \tau^+\tau^{-} $, which has large uncertainties. We note that a calculation of $ J/\psi $ leptonic decay in lattice QCD with fully relativistic charm quarks has been reported in Ref. [52], which shows $ \Gamma(J/\psi\to e^+e^-) = $ 5.48(16) keV and is consist with experimental data. This indicates that the disagreement of our results with data may be caused by the lack of QCD corrections.

      We note that in a recent paper, Soni et al. [53] calculated the quarkonium leptonic decay by using the Cornell potential in a non-relativistic version and with pQCD correction up to NLO, their results for the charmonium cases are neither consistent with the data nor with ours, while their bottomonium results are comparable with ours (see below). While in another paper, Badalian et al. [54] calculated the decay rates for $ \psi(1S-4S) $ with QCD correction at the NLO level by taking the Cornell potential as well as the semi-Salpeter equation, and their results are 5.47 keV, 2.68 keV, 1.97 keV, and 1.58 keV for $ \psi(1S-4S) $ leptonic decays, respectively, which are smaller than our charmonium results. These two comparisons indicate that the relativistic corrections as well as QCD corrections are large for the charmonium system.

    • 6.3.   Bottomonium leptonic decay widths

    • We present the non-relativistic and relativistic results of the bottomonium leptonic decay widths in Table III. Similar to the charmonium case, the relativistic correction is also sizable. For the ground state $ \Upsilon(1S) $, the relativistic effect is about 14%, and for the excited states $ \Upsilon(2S-5S) $, the relativistic effects vary from 20% to 28%. Those predictions agree with the predictions given in the literature. For examples, Bodwin et al. [39] predicted the relativistic effect is 13.2% for $ {v_b}^2\sim0.10 $ by applying the NRQCD up to $ {v_b}^4 $ accuracy; and a Lattice QCD prediction indicated that the relativistic effects are about (15-25)% [22] for $ \Upsilon(1S) $ and $ \Upsilon(2S) $ up to $ {v_b}^2 $ accuracy.

      modesNRRe$ \frac{\rm{NR-Re}}{\rm{Re}} $Exp
      $ \Upsilon(1S)\to e^+e^- $1.47$ ^{+0.23}_{-0.20} $1.29$ ^{+0.19}_{-0.16} $14.0$ ^{+0.9}_{-1.6} $%1.340$ \pm $0.018 (1.29$ \pm $0.09)
      $ \Upsilon(1S)\to\tau^{+}\tau^{-} $1.46$ ^{+0.22}_{-0.20} $1.28$ ^{+0.18}_{-0.16} $14.0$ ^{+1.1}_{-1.5} $%1.40$ \pm $0.09
      $ \Upsilon(2S)\to e^+e^- $0.771$ ^{+0.123}_{-0.125} $0.629$ ^{+0.104}_{-0.088} $22.6$ ^{+0.0}_{-3.2} $%0.612$ \pm $0.011
      $ \Upsilon(2S)\to\tau^{+}\tau^{-} $0.766$ ^{+0.120}_{-0.123} $0.625$ ^{+0.101}_{-0.086} $22.6$ ^{+0.0}_{-3.3} $%0.64$ \pm $0.12
      $ \Upsilon(3S)\to e^+e^- $0.541$ ^{+0.088}_{-0.088} $0.450$ ^{+0.070}_{-0.065} $20.2$ ^{+7.8}_{-2.5} $%0.443$ \pm $0.008
      $ \Upsilon(3S)\to\tau^{+}\tau^{-} $0.538$ ^{+0.086}_{-0.087} $0.448$ ^{+0.068}_{-0.064} $20.2$ ^{+7.7}_{-2.8} $%0.47$ \pm $0.10
      $ \Upsilon(4S)\to e^+e^- $0.429$ ^{+0.083}_{-0.059} $0.355$ ^{+0.058}_{-0.050} $20.8$ ^{+6.6}_{-7.2} $%0.272$ \pm $0.029 (0.322$ \pm $0.056)
      $ \Upsilon(4S)\to\tau^{+}\tau^{-} $0.427$ ^{+0.081}_{-0.057} $0.353$ ^{+0.056}_{-0.050} $20.8$ ^{+6.5}_{-7.1} $%
      $ \Upsilon(5S)\to e^+e^- $0.380$ ^{+0.048}_{-0.069} $0.296$ ^{+0.048}_{-0.038} $28.4$ ^{+1.1}_{-7.9} $%0.31$ \pm $0.07
      $ \Upsilon(5S)\to\tau^{+}\tau^{-} $0.378$ ^{+0.047}_{-0.068} $0.295$ ^{+0.047}_{-0.038} $28.4$ ^{+1.0}_{-7.8} $%

      Table 3.  Decay rates of $ \Upsilon(nS)\to\ell^+\ell^- $ in unit of keV. 'NR' is the non-relativistic result, 'Re' is the relativistic result, 'Exp' means the experimental data [50].

      We should point out that the above large relativistic effect is special for the bottomonium leptonic decays, $ \Upsilon(nS) \to \ell^+\ell^- $, which is not a universal conclusion for processes involving a bottomonium. For the di-lepton decays, its amplitude is proportional to the wave function via a form as $ \int {d^3 \vec{q}} \left[\psi_{5}({\vec q})-\frac{{\vec q}^2}{3M^2}\psi_3(\vec{q})\right] $, i.e. the power of the wave function is one. For other processes, such as the meson A decays to the meson B via a semileptonic way, the corresponding amplitude should be proportional to the overlapping integral of the wave functions for the initial and final states, e.g. $ \int {d^3 \vec{q}}\; \psi_A \cdot \psi_B $. Because the wave functions are large in small q region, comparing with the case of only one wave function, the product of two wave functions will be suppressed in large q region and the contributions from the relativistic terms will be greatly suppressed.

      In Table III, we also give the theoretical uncertainties, which are obtained by varying all the parameters simultaneously within $ \pm 10 $% of the central values, and the largest variations are taken as the errors. Our relativistic results agree well with the experimental data. We also note that, for $ \Upsilon(1S) \to e^+e^- $, the PDG provides us two different data, it directly lists $ \Gamma_{ee} = 1.34 $ keV, but at the same time it also lists the branching ratio $ Br = 2.38 $%, leading to $ \Gamma_{\Upsilon(1S)\to e^+e^-} = 1.29 $ keV with the help of the full width $ \Gamma_{\Upsilon(1S)} = 54.02 $ keV [50]. The second one is the same as our relativistic result. The same thing happens to the case of $ \Upsilon(4S) \to e^+e^- $, the PDG directly lists $ \Gamma_{ee} = 0.272 $ keV [50], but from the branching ratio given in PDG, we get $ \Gamma_{ee} = 0.322 $ keV, and we hope the PDG shall update its data in near future.

      Table III shows that all the relativistic results $ \Gamma_{\ell\ell}(1S-5S) $ are consistent with the experimental data. Our predictions also agree with the Lattice prediction [22], $ \Gamma(\Upsilon(1S)\to e^+e^-) = 1.19(11) $ keV and $ \Gamma(\Upsilon(2S)\to e^+e^-) = 0.69(9) $ keV, the NRQCD prediction [26], $ \Gamma(\Upsilon(1S)\to e^+e^-) = 1.25 $ keV, and the NRQCD prediction with NNNLO pQCD corrections [30], $ \Gamma(\Upsilon(1S)\to e^+e^-) = 1.08\pm0.05(\alpha_s)^{+0.01}_{-0.20}(\mu) $ keV.

    • 6.4.   Lepton flavor university

    • To test the lepton flavor university, we give the ratios of $ R^{\psi_{nS}}_{\tau\tau} $ and $ R^{\Upsilon_{nS}}_{\tau\tau} $ in Table IV, their definitions are similar, for example,

      $ R^{\psi_{2S}}_{\tau\tau} $$ R^{\psi_{3S}}_{\tau\tau} $$ R^{\psi_{4S}}_{\tau\tau} $$ R^{\psi_{5S}}_{\tau\tau} $
      Ours0.391$ ^{+0.210}_{-0.391} $0.668$ ^{+0.073}_{-0.237} $0.767$ ^{+0.026}_{-0.112} $0.819$ ^{+0.039}_{-0.091} $
      $ R^{\Upsilon_{1S}}_{\tau\tau} $$ R^{\Upsilon_{2S}}_{\tau\tau} $$ R^{\Upsilon_{3S}}_{\tau\tau} $$ R^{\Upsilon_{4S}}_{\tau\tau} $$ R^{\Upsilon_{5S}}_{\tau\tau} $
      Ours0.992$ ^{+0.001}_{-0.006} $0.994$ ^{+0.003}_{-0.003} $0.996$ ^{+0.002}_{-0.002} $0.995$ ^{+0.001}_{-0.002} $0.997$ ^{+0.000}_{-0.002} $
      CLEO[12]$ 1.02\pm0.07 $$ 1.04\pm0.09 $$ 1.05\pm0.13 $
      BABAR[14]$ 1.005\pm0.035 $
      PDG [50]$ 1.05\pm0.06 $$ 1.04\pm0.20 $$ 1.05\pm0.24 $

      Table 4.  Ratios of $ R^{\psi_{nS}}_{\tau\tau} = \frac{\Gamma(\psi_{nS}\to\tau^{+}\tau^{-})}{\Gamma(\psi_{nS}\to \mu^{+}\mu^{-})} $ and $ R^{\Upsilon_{nS}}_{\tau\tau} $. For the experimental data from PDG [50], we have added the statistical and systematic uncertainties together.

      $ R^{\psi_{nS}}_{\tau\tau} = \frac{\Gamma(\psi_{nS}\to\tau^{+}\tau^{-})}{\Gamma(\psi_{nS}\to \mu^{+}\mu^{-})}. $

      The deviation of the ratio $ R^{\psi_{nS}}_{\tau\tau} $ from the lepton flavor universality will indicate the presence of new physics beyond the Standard Model.

      Table IV shows the ratios calculated with the 'Re' values. The uncertainty in the ratio comes from the variation of the input parameters. For charmonium cases, the ratios $ R^{\psi_{nS}}_{\tau\tau} $ are quite different from each other, the reason is that the charmonium mass is just a little heavier than those of two τ, and for the same reason, we get a large uncertainty. While for bottomonium ratios $ R^{\Upsilon_{nS}}_{\tau\tau} $, because the τ mass is much smaller than bottomonium mass, we get almost the same value for all the ratios $ R^{\Upsilon_{nS}}_{\tau\tau} $, whose uncertainty is also very small due to the cancellation between the numerator and denominator. Even though, all the central values of the ratios $ R^{\Upsilon_{nS}}_{\tau\tau} $ are smaller than 1, they are consistent with the existing experimental data within errors.

      To cancel the model dependence of the theoretical predictions, we give another two tables, Table V and Table VI, where we show the ratios of $ \Gamma(\psi(nS)\to e^+e^-)/ $$ \Gamma(J/\psi\to e^+e^-) $ and $ \Gamma(\Upsilon(nS)\to \ell^+\ell^-)/\Gamma(\Upsilon(1S)\to \ell^+\ell^-) $, respectively. In case of $ \Upsilon(nS) $ decay, we obtain the same central values for e and τ final states, so we only present the ratio $ \Gamma(\Upsilon(nS)\to \ell^+\ell^-)/\Gamma(\Upsilon(1S)\to \ell^+\ell^-) $ in Table V and Table VI, which are calculated by using the $ e^+e^- $ final states listed in Table III.

      $ \frac{\Gamma(\psi(2S))}{\Gamma(J/\psi)} $$ \frac{\Gamma(\psi(3S))}{\Gamma(J/\psi)} $$ \frac{\Gamma(\psi(4S))}{\Gamma(J/\psi)} $$ \frac{\Gamma(\psi(5S))}{\Gamma(J/\psi)} $
      Ours0.495$ ^{+0.019}_{-0.017} $0.340$ ^{+0.016}_{-0.017} $0.259$ ^{+0.013}_{-0.016} $0.210$ ^{+0.013}_{-0.016} $
      Exp [50]0.42±0.020.15±0.020.086±0.0420.10±0.02

      Table 5.  Ratio of $ \Gamma(\psi(nS)\to e^+e^-)/\Gamma(J/\psi\to e^+e^-) $.

      $ \frac{\Gamma(\Upsilon(2S))}{\Gamma(\Upsilon(1S))} $$ \frac{\Gamma(\Upsilon(3S))}{\Gamma(\Upsilon(1S))} $$ \frac{\Gamma(\Upsilon(4S))}{\Gamma(\Upsilon(1S))} $$ \frac{\Gamma(\Upsilon(5S))}{\Gamma(\Upsilon(1S))} $
      Ours0.488$ ^{+0.008}_{-0.009} $0.349$ ^{+0.003}_{-0.008} $0.275$ ^{+0.004}_{-0.000} $0.229$ ^{+0.003}_{-0.001} $
      Exp1 [50]0.457±0.0140.33±0.010.203±0.024 (0.240±0.045)0.23±0.06
      Exp2 [50]0.47±0.040.34±0.030.21±0.04(0.25±0.06)0.24±0.07

      Table 6.  Ratio of $ \Gamma(\Upsilon(nS)\to \ell^+\ell^-)/\Gamma(\Upsilon(1S)\to \ell^+\ell^-) $. 'Exp1' is the experimental data where $ \Gamma_{ee}(1S) = 1.340\pm0.018 $ keV is used, 'Exp2' is also experimental data but $ \Gamma_{ee}(1S) = 1.29\pm0.09 $ keV. For $ \Upsilon(4S) $, $ \Gamma_{ee}(4S) = 0.272\pm0.029\; (0.322\pm0.056) $ keV is chosen for the result inside (outside) the bracket.

      Table V shows the ratio $ \frac{\Gamma(\psi(2S))}{\Gamma(J/\psi)} $ is larger but close to the experimental data, and the ratios for highly excited states are much larger than the experimental data. Table VI shows the bottomonium leptonic decay ratios. In row of 'Exp1', the value of $ \Gamma_{ee}(1S) = 1.340\pm0.018 $ keV is used, which is directly listed in the PDG. In row of 'Exp2', $ \Gamma_{ee}(1S) = 1.29\pm0.09 $ keV is used, which is calculated by using the branching ratio of $ \Upsilon(1S) \to e^+ e^- $ given in PDG. The same thing happens to $ \Upsilon(4S) $, for the results outside the brackets, $ \Gamma_{ee}(4S) = 0.272\pm0.029 $ keV is taken from the PDG; while for the results inside the brackets, $ \Gamma_{ee}(4S) = 0.322\pm0.056 $ keV is taken from the PDG branching ratio. It is found that all of our theoretical predictions are consistent with the experimental data.

    7.   Summary
    • In the present paper, we have studied the leptonic decays of heavy vector quarkonia. For charmonium decays, not all of its states are consistent with the data, while for the bottomonium decays, almost all of its S-wave states are in good agreement with the data.

      Theoretical results of the ratios of $ \Gamma(\psi(nS)\to e^+e^-)/$$ \Gamma(J/\psi\to e^+e^-) $ and $ \Gamma(\Upsilon(nS)\to \ell^+\ell^-)/\Gamma(\Upsilon(1S)\to \ell^+\ell^-) $ have been given in Ref. [55], where the authors used the potential model, including the $ v_Q^2 $ relativistic corrections and pQCD corrections at NLO. Their results are comparable with us, i.e. the charmonium leptonic decay widths are inconsistent with the data and the bottomonium leptonic widths are in good agreement with the data. This conclusion has also be observed in Ref. [56]. It seems that the same theoretical tool is hard to provide us satisfying results for both the charmonium and bottomonium systems [57], in this paper, there are several possible reasons to cause this difference, may be the instantaneous approximation is good for bottomonium, but not good enough for charmnium, the improvement of Cornell potential may be needed, more important, the perturbative QCD corrections may have larger corrections in charmonium decays than in bottomonium cases. Since the BS equation is an integral equation, so the QCD corrections from gluon ladder diagrams already been included, other kind of QCD corrections to the kernel or to the quark propogators are welcomed in the future to improve the calculations.

      The Bethe-Salpeter method could provide a strict way to deal with the relativistic effect. Within this framework, we have found that the relativistic corrections are large and important for the leptonic decays $ \psi(nS) \to \ell^+\ell^- $ and $ \Upsilon(nS) \to \ell^+\ell^- $. For the case of $ \psi(1S-5S) $ leptonic decays, the relativistic effects are $ 22^{+3}_{-2} $%, $ 34^{+5}_{-5} $%, $ 41^{+6}_{-6} $%, $ 52^{+11}_{-13} $% and $ 62^{+14}_{-12} $%, respectively. So for the highly excited states $ \psi(nS) $, the relativistic corrections give dominant contributions. For the $ \Upsilon(1S-5S) $ decays, the relativistic effects are $ 14^{+1}_{-2} $%, $ 23^{+0}_{-3} $%, $ 20^{+8}_{-2} $%, $ 21^{+6}_{-7} $% and $ 28^{+2}_{-7} $%, respectively. Thus the relativistic effects should be considered for a sound prediction of the heavy quarkonium decays.

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