 Meson hyperfine mass splitting

In a simple constituent quark model, a spin-dependent contribution $A{\mathbf{S}}_1\cdot {\mathbf{S}}_2/(m_1{m}_2)$ splits a spin-zero and spin-one meson having the same spatial configuration. For two spin-half constituents, addition of angular momentum gives ${\mathbf{S}}_1\cdot {\mathbf{S}}_2=[S(S+1)-3/2]/2$, equal to $-3/4$ for $S=0$ and $+1/4$ for $S=1$. ThusThe spin-averaged mass is $(M_0+3M_1)/4=M_{\mathrm{base}}$. Positive $A$ places the vector state above the pseudoscalar. Determining $A$ from just $M_0$ needs an additional assumption about $M_{\mathrm{base}}$; setting it equal to $m_1+m_2$ neglects all other spin-independent binding terms. A splitting below the pion mass closes a one-pion strong decay to the lower meson, often making the allowed electromagnetic transition with photon emission dominant.