Simple Baryon-Meson Mass Relations For A Logarithmic Interquark Potential

I consider the quantity delta(m_1m_2m_3) = M_{q_1q_2q_3} - [M_{q_1q_2bar} + M_{q_2q_3bar} + M_{q_1q_3bar}]/2, where the M's represent the ground state spin-averaged hadron masses with the indicated quark content and the m's the corresponding constituent quark masses. I assume a logarithmic interquark potential, the validity of a nonrelativistic approach, and various standard potential model inputs. Simple scaling arguments then imply that the quantity R(x)=delta(mmm_3)/delta(m_0m_0m_0) depends only on the ratio x=m/m_3, and is independent of m_0 as well as any parameters appearing in the potential. A simple and accurate analytic determination of delta(mmm_3), and hence R(x), is given using the 1/D expansion where D is the number of spatial dimensions. When applicable, this estimate of R(x) compares very well to experiment -- even for hadrons containing light quarks. A prediction of the above result which is likely to be tested in the near future is M_{Sigma_b^*}/2 + [M_{Lambda_b} + M_{Sigma_b}]/4 = 5774 +/- 4 MeV/c^2.