Gell-Mann–Nishijima formula

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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:

$Q = I_z + \frac{1}{2} (B+S)$ ^[1]^[2]

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.