= Meson hyperfine mass splitting
{title2=$M_1-M_0=A/(m_1m_2)$}
In a simple constituent <quark model>, a spin-dependent contribution $A\mathbf S_1\cdot\mathbf S_2/(m_1m_2)$ 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 $S=1$. Thus
$$
M_0=M_{\rm base}-\frac{3A}{4m_1m_2},\qquad M_1=M_{\rm base}+\frac{A}{4m_1m_2},\qquad M_1-M_0=\frac{A}{m_1m_2}.
$$
The spin-averaged mass is $(M_0+3M_1)/4=M_{\rm base}$. Positive $A$ places the vector state above the pseudoscalar. Determining $A$ from just $M_0$ needs an additional assumption about $M_{\rm base}$; 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.
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