Gell-Mann–Nishijima Formula
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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 In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. Isospin is also known as isobaric spin or isotopic spin. Isospin symmetry is a subset of the flavour symmetr ...
''I3'' of quarks and
hadron In particle physics, a hadron is a composite subatomic particle made of two or more quarks held together by the strong nuclear force. Pronounced , the name is derived . They are analogous to molecules, which are held together by the electri ...
s to the
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
''Q''. It was originally given by Kazuhiko Nishijima and Tadao Nakano in 1953, and led to the proposal of strangeness as a concept, which Nishijima originally called "eta-charge" after the eta meson.
Murray Gell-Mann Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American theoretical physicist who played a preeminent role in the development of the theory of elementary particles. Gell-Mann introduced the concept of quarks as the funda ...
proposed the formula independently in 1956. The modern version of the formula relates all
flavour quantum numbers In particle physics, flavour or flavor refers to the ''species'' of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with ''flavour quantum numbe ...
(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 (B+S)\ This equation was originally based on empirical experiments. It is now understood as a result of the
quark model In particle physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks that give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the Eig ...
. In particular, the
electric charge Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
''Q'' of a quark or hadron particle is related to its
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. Isospin is also known as isobaric spin or isotopic spin. Isospin symmetry is a subset of the flavour symmetr ...
''I3'' and its
hypercharge In particle physics, the hypercharge (a portmanteau of hyperonic and charge (physics), charge) ''Y'' of a subatomic particle, particle is a quantum number conserved under the strong interaction. The concept of hypercharge provides a single charg ...
''Y'' via the relation: :Q = I_3 + \frac Y\ :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: :Q = I_3 + \frac (B+S+C+B^\prime+T) where ''Q'' is the charge, ''I''3 the 3rd-component of the
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. Isospin is also known as isobaric spin or isotopic spin. Isospin symmetry is a subset of the flavour symmetr ...
, ''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: :\begin Q &= \frac\left left(n_\text - n_\bar\right) + \left(n_\text - n_\bar\right) + \left(n_\text - n_\bar\right)\right- \frac\left left(n_\text - n_\bar\right) + \left(n_\text - n_\bar\right) + \left(n_\text - n_\bar\right)\right\\ B &= \frac\left[\left(n_\text - n_\bar\right) + \left(n_\text - n_\bar\right) + \left(n_\text - n_\bar\right) + \left(n_\text - n_\bar\right) + \left(n_\text - n_\bar\right) + \left(n_\text - n_\bar\right)\right] \\ I_3 &= \frac[(n_\text-n_\bar)-(n_\text-n_\bar)] \\ S &= -\left(n_\text - n_\bar\right);\quad C = +\left(n_\text - n_\bar\right);\quad B^\prime = -\left(n_\text - n_\bar\right);\quad T = +\left(n_\text - n_\bar\right) \end 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 In theoretical physics, quantum chromodynamics (QCD) is the study of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of ...
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

In 1961 Glashow proposed a relation similar formula would also apply to the
weak interaction In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is th ...
: Q = T_3 + \fracY. Here the charge Q is related to the projection of weak isospin T_3 and the
hypercharge In particle physics, the hypercharge (a portmanteau of hyperonic and charge (physics), charge) ''Y'' of a subatomic particle, particle is a quantum number conserved under the strong interaction. The concept of hypercharge provides a single charg ...
Y.


References


Further reading

* {{DEFAULTSORT:Gell-Mann-Nishijima Formula Standard Model Murray Gell-Mann he:נוסחת גל-מן-נישיג'ימה