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.
Categories: hep-ph