Baryon hyperfine mass formula ($M={\sum }_i{m}_i+A^{\prime }{\sum }_{i<j}⟨{\mathbf{S}}_i\cdot {\mathbf{S}}_j⟩/(m_i{m}_j)$)

This effective baryon mass model adds a colour-magnetic spin-spin interaction to a sum of constituent quark masses. The common coefficient $A^{\prime }$ absorbs a spatial contact density, colour factors, and the strong coupling. $A^{\prime }$ has dimensions of energy cubed in natural units. A shared coefficient is an approximation because different flavour configurations can have different spatial wavefunctions.

Use dimensionless spin angular momentum operators with $\hslash =1$ and the common-coefficient baryon hyperfine mass formula. The identical pair has total spin one, so its pair expectation is $1/4$. For total baryon spin $1/2$, the two mixed-flavour pair expectations sum to $-1$; for total spin $3/2$, they sum to $1/2$. Their difference is $3/2$, giving the title formula for either repeated-flavour choice. The equality assumes the same effective coefficient $A^{\prime }$ in the two flavour configurations; different spatial overlaps can change it.