We derive an assumption-free mass relation that connects heavy-quark mesons (\(c\bar{c}\), \(b\bar{b}\) and \(B_c\) mesons) with different quantum numbers, but same orbital angular momentum. The obtained formula is exact, provided the hyperfine mass splitting, \(\delta _{hf}\), and \(\delta _\xi :=M(n^1 L_\ell )/M(n^3 L_\ell )-1\) are determined independently. For completeness, we obtain the heavy-quark mesons spectrum following a Schrödinger wave equation (SWE) approach. The Song-Lin potential is employed along with the corresponding relativistic corrections, namely: spin-spin, spin-orbit and tensor interactions. Our numerical strategy is based upon the Numerov method, which enables us to cast our SWE approach into a matrix eigenvalue problem, for which high numerical accuracy can be achieved. Notably, only a single set of parameters is required to produce the known spectrum of the heavy-quark mesons.

我们推导出一种无假设的质量关系，该关系连接具有不同量子数的重夸克介子（\(c\bar{c}\)、\(b\bar{b}\)和\(B_c\)介子），但是相同的轨道角动量。得到的公式是准确的，前提是超精细质量分裂\(\delta _{hf}\)和\(\delta _\xi :=M(n^1 L_\ell )/M(n^3 L_\嗯)-1\)是独立确定的。为完整起见，我们按照薛定谔波动方程 (SWE) 方法获得重夸克介子谱。松林势与相应的相对论修正一起使用，即：自旋-自旋、自旋-轨道和张量相互作用。我们的数值策略基于 Numerov 方法，这使我们能够将 SWE 方法转化为矩阵特征值问题，从而可以实现高数值精度。值得注意的是，只需要一组参数就可以产生已知的重夸克介子谱。