= Quark-model constituent bookkeeping
{title2=$Q=\sum_i Q_i,\quad B=\sum_i B_i,\quad S=\sum_i S_i$}
For a three-light-flavor example, take <up quarks>, <down quarks> and <strange quarks> with the following additive <quantum numbers>. Here $Q$ denotes <electric charge> in units of the positive <elementary charge> $e$, $B$ is <baryon number>, $S$ is <strangeness>, $I_3$ is the third component of <isospin>, and $Y=B+S$ is <flavor hypercharge>. An <antiquark> reverses all five displayed values. These are flavor coordinates, not spatial coordinates or <electroweak hypercharge>.
|| <Quark flavor>
|| $Q$
|| $B$
|| $S$
|| $I_3$
|| $Y$
| $u$
| $2/3$
| $1/3$
| $0$
| $1/2$
| $1/3$
| $d$
| $-1/3$
| $1/3$
| $0$
| $-1/2$
| $1/3$
| $s$
| $-1/3$
| $1/3$
| $-1$
| $0$
| $-2/3$
Adding these columns immediately explains familiar <hadrons>. A <proton> with <valence quarks> $uud$ has $Q=2/3+2/3-1/3=1$ and $B=1$; a <neutron> with $udd$ has $Q=0$ and $B=1$. A positive <kaon> with $u\bar s$ has $Q=2/3+1/3=1$, $B=0$ and $S=+1$. Thus the same constituent rules describe both <baryons> and <mesons>.
\Image[/quark-charge-bookkeeping-and-neutral-states.png]
{title=Adding constituent <quantum numbers> distinguishes charged and neutral <hadrons>}
{description=Original arithmetic examples for the three-light-flavor <quark model>. The lower panel contrasts two orthogonal neutral flavor combinations: identical total <electric charge> does not make two states identical. These are ideal flavor basis states, not a mass fit or a reconstruction of an experimental event.}
Two <hadrons> can have identical additive labels while differing in their internal state. The <Lambda baryon> and neutral <Sigma baryon> both have valence content $uds$, yet their total <isospin> differs. Likewise, the neutral <pion> flavor direction $(u\bar u-d\bar d)/\sqrt2$ and the <Eta octet state> $(u\bar u+d\bar d-2s\bar s)/\sqrt6$ both have $Q=Y=0$, but their <inner product> vanishes. Expanding that <inner product> gives $(1-1+0)/\sqrt{12}=0$. Constituent counting must therefore be supplemented by <quantum superposition> and the <symmetry> of the state.
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