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Gell-Mann–Nishijima formula

The original form of the Gell-Mann–Nishijima formula is:

${\displaystyle Q=I_{3}+{\frac {1}{2}}(B+S)\ }$

This equation was originally based on empirical experiments. It is now understood as a result of the quark model. In particular, the electric charge Q of a quark or hadron particle is related to its isospin I₃ and its hypercharge Y via the relation:

${\displaystyle Q=I_{3}+{\frac {1}{2}}Y\ }$

${\displaystyle Y=2(Q-I_{3})}$

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

${\displaystyle Q=I_{3}+{\frac {1}{2}}(B+S+C+B^{\prime }+T)}$

where Q is the charge, I₃ the 3rd-component of the isospin, B the baryon number, and S, C, B′, T are the strangeness, charm, bottomness and topness numbers.

Expressed in terms of quark content, these would become:

${\displaystyle {\begin{aligned}Q&={\frac {2}{3}}\left[\left(n_{\text{u}}-n_{\bar {\text{u}}}\right)+\left(n_{\text{c}}-n_{\bar {\text{c}}}\right)+\left(n_{\text{t}}-n_{\bar {\text{t}}}\right)\right]-{\frac {1}{3}}\left[\left(n_{\text{d}}-n_{\bar {\text{d}}}\right)+\left(n_{\text{s}}-n_{\bar {\text{s}}}\right)+\left(n_{\text{b}}-n_{\bar {\text{b}}}\right)\right]\\B&={\frac {1}{3}}\left[\left(n_{\text{u}}-n_{\bar {\text{u}}}\right)+\left(n_{\text{c}}-n_{\bar {\text{c}}}\right)+\left(n_{\text{t}}-n_{\bar {\text{t}}}\right)+\left(n_{\text{d}}-n_{\bar {\text{d}}}\right)+\left(n_{\text{s}}-n_{\bar {\text{s}}}\right)+\left(n_{\text{b}}-n_{\bar {\text{b}}}\right)\right]\\I_{3}&={\frac {1}{2}}[(n_{\text{u}}-n_{\bar {\text{u}}})-(n_{\text{d}}-n_{\bar {\text{d}}})]\\S&=-\left(n_{\text{s}}-n_{\bar {\text{s}}}\right);\quad C=+\left(n_{\text{c}}-n_{\bar {\text{c}}}\right);\quad B^{\prime }=-\left(n_{\text{b}}-n_{\bar {\text{b}}}\right);\quad T=+\left(n_{\text{t}}-n_{\bar {\text{t}}}\right)\end{aligned}}}$

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.

From a quantum chromodynamics point of view, the Gell-Mann–Nishijima formula and its generalized version can be derived using an approximate SU(3) flavour symmetry because the charges can be defined using the corresponding conserved Noether currents.

Weak interaction analog[edit]

In 1961 Glashow proposed a relation similar formula would also apply to the weak interaction:^{[4][5]: 152} $${\displaystyle Q=T_{3}+{\frac {1}{2}}Y.}$$ Here the charge ${\displaystyle Q}$ is related to the projection of weak isospin ${\displaystyle T_{3}}$ and the hypercharge ${\displaystyle Y}$.

References[edit]

Further reading[edit]

• Griffiths, DJ (2008). Introduction to Elementary Particles (2nd ed.). Wiley-VCH. ISBN 978-3-527-40601-2.