Gell-Mann–Nishijima formula

The Gell-Mann–Nishijima formula relates the baryon number B, the strangeness S, the isospin I_z of hadrons to the charge Q. It was originally, in 1955 and 1956, given as:

$Q\; =\; I\_z\; +\; frac\{1\}\{2\}\; (B+S)$

Originally, this equation was based on empirical experiments. It is now understood as a result of the Quark model. In particular, the electric charge Q of a particle is related to its isospin I_z and its hypercharge Y via the relation:

$Q\; =\; I\_z\; +\; frac\{1\}\{2\}\; Y$

Since the discovery of charm, top, and bottom quark flavors, this formula has been generalized . It now takes the form:

$Q\; =\; I\_z\; +\; frac\{1\}\{2\}\; (B+S+C+B^prime+T)$

where Q is the charge, I_z the z-component of the isospin, B the baryon number, and S, C, B′, T are the strangeness, charmness, bottomness and topness numbers.

Expressed in terms of quark content, these would become:
$Q=frac\{2\}\{3\}(n\_u-n\_bar\{u\}+n\_c-n\_bar\{c\}+n\_t-n\_bar\{t\})-frac\{1\}\{3\}(n\_d-n\_bar\{d\}+n\_s-n\_bar\{s\}+n\_b-n\_bar\{b\})$
$B=frac\{1\}\{3\}(n\_u-n\_bar\{u\}+n\_c-n\_bar\{c\}+n\_t-n\_bar\{t\}+n\_d-n\_bar\{d\}+n\_s-n\_bar\{s\}+n\_b-n\_bar\{b\})$
$I\_z=frac\{1\}\{2\}[((n\_u-n\_bar\{u\})-(n\_d-n\_bar\{d\})]$
$S=-frac\{1\}\{2\}((n\_s-n\_bar\{s\}),\; C=+frac\{1\}\{2\}((n\_c-n\_bar\{c\}),\; B^prime=-frac\{1\}\{2\}((n\_b-n\_bar\{b\}),\; T=+frac\{1\}\{2\}((n\_t-n\_bar\{t\})$

By convention, the flavor quantum numbers, strangeness, charm, bottomness, and topness carry the same sign as the electric charge of the particle. So, since the strange and bottom quarks have a negative charge, they have flavor quantum numbers equal to -1. And Since the charm and top quarks have positive electric charge, their flavor quantum numbers are +1.

References

Introduction to Elementary Particles by David Griffiths, Wiley .