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The Gell-Mann–Nishijima formula (sometimes known as the NNG formula) relates the baryon number B, the strangeness S, the isospin I₃ of hadrons to the charge Q. It was originally given by Kazuhiko Nishijima and Tadao Nakano in 1953,^[1] and lead to the proposal of strangeness as a concept, which Nishijima originally called "eta-charge" after the eta meson.^[2] Murray Gell-Mann proposed the formula independently in 1956.^[3] The modern version of the formula relates all flavour quantum numbers (isospin up and down, strangeness, charm, bottomness, and topness) with the baryon number and the electric charge.

[edit] Formula

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

$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 particle is related to its isospin I₃ and its hypercharge Y via the relation:

$Q = I_3 + \frac{1}{2} Y.\$

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

$Q = I_3 + \frac{1}{2} (B+S+C+B^\prime+T)$

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:

$Q = \frac{2}{3}\left[\left(n_\text{u} - n_\bar{\text{u}}\right) + \left(n_\text{c} - n_\bar{\text{c}}\right) + \left(n_\text{t} - n_\bar{\text{t}}\right)\right] - \frac{1}{3}\left[\left(n_\text{d} - n_\bar{\text{d}}\right) + \left(n_\text{s} - n_\bar{\text{s}}\right) + \left(n_\text{b} - n_\bar{\text{b}}\right)\right]$

$B = \frac{1}{3}\left[\left(n_\text{u} - n_\bar{\text{u}}\right) + \left(n_\text{c} - n_\bar{\text{c}}\right) + \left(n_\text{t} - n_\bar{\text{t}}\right) + \left(n_\text{d} - n_\bar{\text{d}}\right) + \left(n_\text{s} - n_\bar{\text{s}}\right) + \left(n_\text{b} - n_\bar{\text{b}}\right)\right]$

$I_3=\frac{1}{2}[(n_\text{u}-n_\bar{\text{u}})-(n_\text{d}-n_\bar{\text{d}})]$

$S=-\left(n_\text{s}-n_\bar{\text{s}}\right);$ $C=+\left(n_\text{c}-n_\bar{\text{c}}\right);$ $B^\prime=-\left(n_\text{b}-n_\bar{\text{b}}\right);$ $T=+\left(n_\text{t}-n_\bar{\text{t}}\right)$

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.

[edit] References

^ Nakano, T; Nishijima, N (1955). "Charge Independence for V-particles". Progress of Theoretical Physics 10 (5): 581. doi:10.1143/PTP.10.581.
^ Nishijima, K (1955). "Charge Independence Theory of V Particles". Progress of Theoretical Physics 13 (3): 285. doi:10.1143/PTP.13.285.
^ Gell-Mann, M (1956). "The Interpretation of the New Particles as Displaced Charged Multiplets". Il Nuovo Cimento 4: 848. doi:10.1007/BF02748000.

[edit] Further reading

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

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[0] Nakano, T; Nishijima, N (1955). "Charge Independence for V-particles". Progress of Theoretical Physics 10 (5): 581. doi:10.1143/PTP.10.581.

[1] Nishijima, K (1955). "Charge Independence Theory of V Particles". Progress of Theoretical Physics 13 (3): 285. doi:10.1143/PTP.13.285.

[2] Gell-Mann, M (1956). "The Interpretation of the New Particles as Displaced Charged Multiplets". Il Nuovo Cimento 4: 848. doi:10.1007/BF02748000.

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