Here are some quark models of *a*-mesons. All *a*-mesons are built-up around the heart of familial seeds shown on the left. Particles with a different spin or charge are modeled by attaching various quarks around this common kernel. Then excited states are obtained by adding even more quarks. The *a*-mesons may share other quarks in common, beyond these familial ones. But this nugget is the minimum necessary to distinguish the *a*-mesons from other nuclear particle families. Particles are classified on this basis. WikiMechanics analyzes the physics of *a*-mesons using chains of events noted by $\Psi = \left( \sf{\Omega}_{1}, \sf{\Omega}_{2}, \sf{\Omega}_{3} \ \ldots \ \right)$ where each repeated cycle $\, \sf{\Omega} \,$ is composed of the following quarks.

All the foregoing quark models perfectly replicate the quantum numbers of *a*-mesons. They also accurately represent observed values for lifetimes, widths and the mass. With very few exceptions, they are within experimental uncertainty. But with so many quarks it is difficult to see how the models work. So to view the underlying pattern, we remove most of the quark/anti-quark pairs. These $\begin{align} \sf { q \overline{q} } \end{align}$ pairs are needed for equilibrium. Without them, many excited states are so unstable that they are not observed. But the field of $\begin{align} \sf { q \overline{q} } \end{align}$ pairs obscures the minimum number of quarks required to identify a particle and account for its mass. These minima are called **core** coefficients. They show more clearly how excited *a*-mesons are built-up over blocks of the same baryonic quarks. The mass depends on $\Delta n$ not $n$, so $m$ is unchanged by any variation in the field of $\begin{align} \sf { q \overline{q} } \end{align}$ pairs. A particle's rest mass is completely determined by its core quarks.

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