Mass Spectra of Heavy Quarkonia and B_(c) Decay Constant for Static Scalar-Vector Interactions with Relativistic Kinematics

We reproduce masses of the self-conjugate and non-self-conjugate mesons in the context of the spinless Salpeter equation taking into account the relativistic kinematics and the quark spins. The hyperfine splittings for the ${\rm 2S}$ charmonium and ${\rm 1S}$ bottomonium are also calculated. Further, the pseudoscalar and vector decay constants of the $B_{c}$ meson and the unperturbed radial wave function at the origin are also calculated. We have obtained a local equation with a complete relativistic corrections to a class of three attractive static interaction potentials of the general form $V(r)=-Ar^{-\beta}+\kappa r^{\beta}+V_{0},$ with $\beta =1,1/2,~3/4$ decomposed into scalar and vector parts in the form $V_{V}(r)=-Ar^{-\beta }+(1-\epsilon)\kappa r^{\beta}$ and $V_{S}(r)=\epsilon \kappa r^{\beta }+V_{0};$ where $0\leq \epsilon \leq 1.$ We have used the shifted large-N-expansion technique (SLNET) to solve the reduced equation for the scalar $(\epsilon =1),$ equal mixture of scalar-vector $(\epsilon =1/2),$ and vector $(\epsilon =0)$ confinement interaction kernels. The energy eigenvalues are carried out up to the third order approximation.