Wave Functions and Leptonic Decays of Bottom Mesons In the Relativistic Potential Model

We study the wave functions and purely leptonic decays of $b$-flavored mesons (pseudoscalars, vector mesons, and higher excited states that are well established in experiment) in the relativistic potential model based on our previous works. The wave functions are obtained by solving the wave equation including the spin-spin and spin-orbit corrections in the effective potential. The decay constants of $B$, $B^*$ and some of their excited states that have been found in experiment are calculated with these wave functions. Then the branching fractions of the purely leptonic decay modes of these bottom mesons are studied. Our results are in well agreement with experiment data for decay modes that have been measured in experiments. We also provide predictions for some yet unmeasured channels, which are useful for experimental test in the future.

Compared with our previous work [6] where only decay constants of pseudoscalar B and B s mesons are considered, here we extend our earlier work by including the decay constants and pure leptonic decays of vector and higher excited states of bottom mesons.
We upgrade the scenario for treating the energy-momentum conservation for the quarkantiquark inside the bottom mesons. The theoretical results for the leptonic decays are compared with experimental data. For the measured decay modes, our prediction are well consistent with experiment. For the yet unmeasured decay modes, our prediction could be useful for experimental test in the future.
The paper is organized as followings. In Section II, we briefly present the theoretical framework for relativistic potential model, give the solved wave function for the bottom mesons. The formulas to calculate the decay constants and branching ratios are also given here. Section III is devoted to numerical results and discussions. Finally, Section IV is for the conclusion and summary.

II. THEORETICAL FRAMEWORK
A. Relativistic potential model and bound-state wave functions The heavy-light quark-antiquark bound-state systems have been extensively studied with the relativistic potential model in our previous works [6][7][8][9]. The bound state wave functions of mesons can be obtained by solving a Schrödinger type equation where H 0 + H is the effective Hamiltonian, which can be found in Ref. [9] and E is meson's energy. The term H 0 reads, with where V (r) is the effective potential for the strong-interaction between the quark and antiquark [10][11][12]. The first term − 4 3 α S (r) r in V (r) originates from the one-gluon-exchange diagram for the short distance contributions, and br is for confinement effects in long distance, while c is a phenomenological parameter for this heavy-light quark-antiquark system. The other term H contains spin-spin hyper-fine interactions and spin-orbit interactions, which are not given explicitly here (see Ref. [9]).
Using the method described in [7] and developed in [8,9], the wave equation in Eq. (1) can be solved numerically. The wave functions in momentum space can be written as, where the subscripts nlm stands for n-th radial wave function (n = 1 is the lowest), l orbital angular momentum quantum number (l = 0, 1, 2, . . . ), and m the magnetic quantum number corresponding to l. ϕ nl (k) is the radial wave functions and Y lm (k) is the spherical harmonics.
The normalization condition for the wave function is The details for solving the wave equation can be found in our previous works in Refs. [7][8][9], which will not be given here for briefness. By solving the wave equation numerically, the wave function can be obtained. In practice, it is convenient to give an analytical form for radial wave functions by fitting the numerical solution. We find the wave function can be fitted with the following exponential form, Next, we give the obtained results for the parameters a 1 ∼ a 4 for each quantum states.  Mesons (1) For pseudoscalars J P = 0 − , we ignore the difference between the light quark masses m u and m d , therefore the radial wave functions of B ± and B d shall be the same. The wave functions of B ± /B d , B s , and B ± c are depicted in FIG. 1. It is noted that the differences between B ± /B d and B s is relatively smaller than that between B ± /B d and B ± c , since the mass of c quark is greatly larger than that of the light-quarks. The results for the parameters a 1 , a 2 , a 3 , a 4 we obtained are listed in Table I, (2) For vector mesons J P = 1 − , in our model [8,9], they are mixing-states of S-wave (l = 0) eigenstate | 3 S 1 and D-wave (l = 2) eigenstate | 3 D 1 . The S-wave and D-wave wave functions should be given separately. We use Ψ S V (    Mesons collected in Table II.

B. Decay constants
Using the wave function of the quark-antiquark bound-state, the state of a bottom meson can be written as [7,13] |M ( #" (7) where X Sms is the spin wave function, a i † #" k Q ,s 1 , b i † #" k q ,s 2 are creation operators, and S, m s , s 1 , s 2 are the corresponding spin-related quantum numbers. The superscript i is color index and 0.5   the normalization factor N L is obtained via the normalization condition of the meson state The anti-commuting relation of the quark and anti-quark annihilation and creation op- The energy and momentum conservation between the meson and its constituent quark and antiquark should hold when considering the decays of the bottom mesons. We take , and P = (m P , #" 0 ) as the four-momenta of the light quark, the heavy quark and the meson in rest frame, respectively. Due to the energy and momentum conservation, one has To keep the four-momentum conservation, the heavy quark is taken off-shell, while the light quark is kept on-shell in the Altarelli-Cabibbo-Corbo-Maiani-Martinelli (ACCMM) scenario [14,15]. Here we extend the ACCMM scenario by taking both the light and heavy quark off-shell. The off-shell of the quarks are a simple treatment for including the energy and momentum carried by the color field around the quarks. Both the masses of the light and heavy quarks are taken to be running masses The running masses m q (k) and m Q (k) are restricted to be positive in this work. With Eqs. (12) and (13), one can obtain It is not enough to determine the explicit dependence of the running masses m q,Q (k) on the quark momentum k with the above equation. We assume the ratio of m q (k)/m Q (k) is a fixed parameter in this work, i.e., we define the ratio for each quark-antiquark pair as In our numerical treatment in the following, we find that the fixed ratio R i can indeed accommodate the experimental data for the measured leptonic decays of the bottom mesons, and the value of the ratio R i is approximately around the ratio of current masses of the light and heavy quarks m q /m b .
Substituting Eq. (7) into the Eq. (16), we obtain the decay constant For vector mesons B * ± , B * , B * s , there are two types of decay constants that are defined according to different currents where µ is the polarization vector, σ µν = i 2 [γ µ , γ ν ] is the Dirac tensor matrix. Similarly, the decay constants can be obtained as The vector mesons are mixing states of |n 1 3 S 1 and |n 2 3 D 1 with n 1 , n 2 = 1, 2, 3 (details can be found in Refs. [8,9]). The wave function Ψ n00 ( #" k ) in Eqs.  For the leptonic decays of the neutral bottom mesons, the decays are induced by penguin diagrams. The effective Hamiltonian describes such decays is [16][17][18], where λ q = V tb V * tq and the operators are (we use b → s as an example and b → d is similar), where α = e 2 4π is the electromagnetic coupling constant. Except for the contribution of the operators Q 7 and Q 9 , the operators Q 1−6,8 also contribute to the decay process B * 0 d(s) → l + l − up to next-to-leading (NLO) order in α s expension in QCD. The contributions from the operators Q 1−6,8 can be absorbed by a redefinition of two effective Wilson coefficients C 7,9 → C ef f 7,9 . The explicit form of C ef f 7,9 can be found in Refs. [18][19][20], we do not repeat it here.
Next we give the branching ratio of the pure leptonic decay of the neutral bottom mesons.
The branching ratio of B 0 d(s) → l + l − is [21]: The hat over B indicates that it is the averaged time-integrated branching ratio that depends on the details of B 0 q −B 0 q mixing [22]. Γ q H denotes the total decay width of the heavier masseigenstate . In Ref. [21], the authors define a different Wilson coefficient C A and its relation to C 10 in Eq. (21) is straightforward: |C A | = sin 2 θ W 2 |C 10 |, where θ W is the weak-mixing angle (the Weinberg angle).
For the process of the vector meson decaying into charged lepton-anti-lepton pair, the branching ratio reads [23],
The parameters used in this work are collected in Table III, which are quoted from PDG [24].
Our numerical results are listed in Table IV. The uncertainties of branching ratios mainly come from decay constants (whose relative error is 5% in our scenario) and CKM matrix elements (see the full review in paper [24]). For the two well-measured decay modes B + → τ + ν τ and B s → µ + µ − , our results are in good agreement with experiment data. The other predicted branching ratios are all consistent with experimental data. They are well below the experimental upper limit. The branching ratios of B + → µ + ν µ and B d → µ + µ − are very near to the present experimental upper limit, which are very hopeful to be detected with the upgraded detectors at Belle II and/or LHCb [32] in the near future.
In addition, the prediction of the branching ratio of B + c → τ + ν τ is two orders of magnitude larger than that of B + . Therefore it could be a possible channel to be measured in experiments in future.
(2) For vector B mesons J P = 1 − Since the total decay widths of vector B-meson are not well measured in experiments up to now, in this work, their values of theoretical estimations will be used. As stated in Refs. [33][34][35], vector B-mesons' decays are dominated by the electromagnetic processes B * → Bγ and thus we make the assumption that the total decay width Γ approximately equals