The mass relation ${ M_{ 0^{ +}}+3M_{ 1^{ +\prime}}+5M_{ 2^{ +}}= 9M_{ 1^{ +}}}$ miraculously holds for the $P$-wave charmonium $(c\bar{ c})$ and bottomonium $(b\bar{ b})$ systems with soaring precision. The origin of such relation can be addressed from Quark Models, and have been confirmed experimentally in a limited number of cases. In this connection, we propose $M_{ 0^{ +}}+5M_{ 2^{ +}}=3(M_{ 1^{ +\prime}}+M_{ 1^{ +}})$ as an extension to the $P$-wave $B_{ c}$ case. In order to test its applicability, we employ a variety of Quark Model predictions for the $B_c$ mass spectrum. Our numerical analysis confirms such formula is accurate up to very small deviations.
Categories: nucl-th