Mass spectra of heavy pseudoscalars using instantaneous Bethe-Salpeter equation with different kernels

We solved the instantaneous Bethe-Salpeter equation for heavy pseudoscalars in different kernels, where the kernels are obtained using linear scalar potential plus one gluon exchange vector potentials in Feynman gauge, Landau gauge, Coulomb gauge and time-component Coulomb gauge. We obtained the mass spectra of heavy pseudoscalars, and compared the results between different kernels, found that using the same parameters we obtain the smallest mass splitting in time-component Coulomb gauge, the similar largest mass splitting in Feynman and Coulomb gauges, middle size splitting in Landau gauge.

can choose different gauges, so this paper, we will study the mason mass spectra in different kernels caused by different gauges, and see the differences.
In section II, we give a brief review of the Bethe-Salpeter equation and Salpeter equation, the latter is the instantaneous version of the former. In Sec. III, we show how we obtain the vector potentials using the single-gluon exchange in different gauges. In Sec. IV, the relativistic wave function is given. The results of mass spectra of pseudoscalar using different kernels and a short discussion are present in Sec. V.

II. BETHE-SALPETER EQUATION AND SALPETER EQUATION
In this section, we briefly review the Bethe-Salpeter (BS) equation [25], and its instantaneous version, Salpeter equation [26], both of them are the relativistic dynamic equation describing a bound state. Figure 1 show that a quark and an anti-quark are bounded by the kernel V to a meson within the framework of the BS equation, where the index of color is ignored. According to this figure, the BS equation can be written as [25] where χ P (q) is the relativistic wave function of the bound state, V (P, k, q) is the interaction kernel, m 1 , m 2 are the masses of the constitute quark and anti-quark. The relation between the whole momentum P , the relative momentum q, and quark momenta p 1 and p 2 are p 1 = α 1 P + q, p 2 = α 2 P − q, where α 1 = m 1 m 1 +m 2 and α 2 = m 2 m 1 +m 2 . As a four-dimensional integral equation, the BS equation is hard to be solved, so many approximate versions are proposed. Among them, the famous Salpeter equation [26] is the instantaneous approach of BS equation, which is suitable for a bound state with one heavy quark, very good for a doubly heavy quarkonium.
When we take the instantaneous approximation, in the center of mass frame of the bound state, the kernel V (P, k, q) becomes to V (k ⊥ , q ⊥ ), where q µ ⊥ ≡ q µ −q µ , and q µ ≡ (P ·q/M 2 )P µ , M is the mass of meson. We introduce the three-dimensional wave function where q P = (P ·q) M . Then the BS Eq. (1) can be rewritten as where we have defined S 1 (p 1 ) and S 2 (p 2 ) are the propagators where , where i = 1 and 2 for quark and anti-quark, respectively. With the definition Ψ ±± , the wave function can be rewritten as four terms where Ψ ++ P (q ⊥ ) and Ψ −− P (q ⊥ ) are usually called as the positive and negative energy wave functions.
Integrating over q P on both sides of Eq. (3), the Salpeter equation [26] is obtained: The upper Salpeter equation can be also described as four equations by using the project operators: The normalization condition is

III. KERNELS IN DIFFERENT GAUGES
In our calculation, the interaction kernel can be written as V = V s + V v , where V s is the scalar potential, and V v the vector potential. In this paper, the long range linear confining scalar potential λr for V s is chosen, which can not be derived from the perturbative QCD, but suggested by the Color confinement of quarks in bound state and by Lattice QCD calculations [16]. In a potential model, a free constant V 0 is usually introduced in the scalar interaction to fit data, V s = λr + V 0 .
The vector part V v can be derived using the perturbative QCD within the BS equation.
In figure 2, the diagram of the BS equation with one gluon exchange is plotted. According to this Feymann diagram, the BS equation can be written as where α, β, ρ, σ, a and b are the color indices, T a is the SU(3) group generator, and all the repeated indices are summed over. −iD µν δ ab is the gluon propagator, in Feymann gauge (η = 1) or in Landau gauge (η = 0), it can be written as in Coulomb gauge, in time-component Coulomb gauge (also called time-component Lorentz gauge [24]), After we summed over the color index, (T a T a ) σβ = N 2 −1 2N δ σβ = 4 3 δ σβ , the BS equation becomes to where α s = g 2 4π is the running coupling constant, in momentum space and at one-loop level, it can be written as , and e = 2.71828, N f is the active quark number, N f = 3 for charmonium and heavy-light system, N f = 4 for bottomonium.
In order to avoid the infrared divergence in momentum space and incorporate the screening effects, we add an exponential factor e −αr to the potential [27], λr → λ α (1 − e −αr ). In the instantaneous approach, the four dimensional quantity q − k becomes to q ⊥ − k ⊥ . So in the momentum space and the rest frame of the bound state, the scalar potential takes the form: where the radial wave function ϕ i is a function of q 2 ⊥ , so no q 2 ⊥ and higher order terms appear in the wave function. Because of the instantaneous approach P · q ⊥ = 0, also no P · q ⊥ = 0 terms in the wave function. We note that the q dependent terms are relativistic, if we delete them, and set ϕ 1 (q ⊥ ) = ϕ 2 (q ⊥ ), then the wave function reduces to the non-relativistic case.
Not all the four radial wave functions ϕ i are independent, equation Eq.(10) provide the connections .
We have two equations Eqs. (8)(9), and two unknown radial wave functions ϕ 1 (q ⊥ ) and ϕ 2 (q ⊥ ), after taking trace, we can solve the coupled equations as the eigenvalue problem, and obtain the mass spectrum and radial wave functions numerically. In table I, we show the mass spectra of D 0 (nS) using four different kernels, and the experimental data from Particle Data Group [29] are also shown. As shown in Sec.III, the kernels are obtained by using linear scalar potential plus one gluon exchange vector potential, where the gluon propagator has been set in Feynman, Landau, Coulomb, and  The mass spectra of D s , B and B s are given in Tables II-IV The meson B c (2S) is first observed by ATLAS Collaboration [4], and the mass is measured to be 6842 ± 4 ± 5 MeV. Later, CMS Collaboration reported the observation of the B c (2S) and B * c (2S), and the mass peak of these two mesons is located at 6871.2±1.2±0.8±0.8 MeV,      Mass spectra of charmonium and bottomonium are shown in Table VI and Table VII. For charmonium, the first radial excited state η c (2S) has been detected, whose mass is 3637 In summary, we solved the instantaneous BS equation, which is also called Salpeter equation for heavy pseudoscalars, using four different kernels caused by different gauges. We found in the current choice of parameters, the prediction of Landau gauge is more consistent with experimental data. But we should point out that, if another set of parameters is chosen, the conclusion may be different, this is also the reason why different potentials and gauges existing in literature, but the following conclusion is remain unchanged, the mass splitting