Gell-Mann–Nishijima formula

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 led 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.

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.

References

1. Nakano, T; Nishijima, N (1953). "Charge Independence for V-particles". Progress of Theoretical Physics 10 (5): 581. Bibcode:1953PThPh..10..581N. doi:10.1143/PTP.10.581.
2. Nishijima, K (1955). "Charge Independence Theory of V Particles". Progress of Theoretical Physics 13 (3): 285. Bibcode:1955PThPh..13..285N. doi:10.1143/PTP.13.285.
3. Gell-Mann, M (1956). "The Interpretation of the New Particles as Displaced Charged Multiplets". Il Nuovo Cimento 4 (S2): 848. doi:10.1007/BF02748000.

Further reading

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