Solving the Puzzle of ${M_{D^*}-M_{D}\over M_{D_s^*}-M_{D_s}}$ $\simeq$ ${M_{B^*}-M_{B}\over M_{B_s^*}-M_{B_s}}$ $\simeq$ $1$

Abstract

The commonly used Hamiltonian of the chromomagnetic hyperfine splitting is inversely proportional to the product of the masses of two constituent quarks composing the meson. So it is expected to have $(M_{D^*}-M_{D})/(M_{D_s^*}-M_{D_s})$ $\simeq$ $(M_{B^*}-M_{B})/(M_{B_s^*}-M_{B_s})$ $\simeq$ $1.6$, when the constituent quark masses $m_{u,d}=0.33$ GeV and $m_s=0.53$ GeV are used. However, the experimental results show that the above ratios are very close to 1. We solve this puzzle by employing the Hamiltonian recently proposed by Scora and Isgur.