The Gell-Mann–Nishijima formula (sometimes known as the NNG formula) relates the

baryon number In particle physics, the baryon number is a strictly conserved additive quantum number of a system. It is defined as ::B = \frac\left(n_\text - n_\bar\right), where ''n''q is the number of quarks, and ''n'' is the number of antiquarks. Bary ...

''B'', the

strangeness In particle physics, strangeness ("''S''") is a property of particles, expressed as a quantum number, for describing decay of particles in strong and electromagnetic interactions which occur in a short period of time. The strangeness of a parti ...

''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. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...

''I₃'' of

quarks A quark () is a type of elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. All commonly o ...

and

hadron In particle physics, a hadron (; grc, ἁδρός, hadrós; "stout, thick") is a composite subatomic particle made of two or more quarks held together by the strong interaction. They are analogous to molecules that are held together by the ele ...

s to the

electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons respe ...

''Q''. It was originally given by

Kazuhiko Nishijima (4 October 1926 – 15 February 2009) was a Japanese physicist who made significant contributions to particle physics. He was professor emeritus at the University of Tokyo and Kyoto University until his death in 2009. He was born in Tsuchiura ...

and Tadao Nakano in 1953, and led to the proposal of

as a concept, which Nishijima originally called "eta-charge" after the

eta meson The eta () and eta prime meson () are isosinglet mesons made of a mixture of up, down and strange quarks and their antiquarks. The charmed eta meson () and bottom eta meson () are similar forms of quarkonium; they have the same spin and p ...

Murray Gell-Mann Murray Gell-Mann (; September 15, 1929 – May 24, 2019) was an American physicist who received the 1969 Nobel Prize in Physics for his work on the theory of elementary particles. He was the Robert Andrews Millikan Professor of Theoretical ...

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 numbers'' th ...

(isospin up and down, strangeness,

charm Charm may refer to: Social science * Charisma, a person or thing's pronounced ability to attract others * Superficial charm, flattery, telling people what they want to hear Science and technology * Charm quark, a type of elementary particle * Ch ...

bottomness In physics, bottomness (symbol ''B''′ using a prime as plain ''B'' is used already for baryon number) or beauty is a flavour quantum number reflecting the difference between the number of bottom antiquarks (''n'') and the number of bottom q ...

, and

topness Topness (''T'', also called truth), a flavour quantum number, represents the difference between the number of top quarks (t) and number of top antiquarks () that are present in a particle: :T = n_\text - n_\bar By convention, top quarks have a ...

) 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 which give rise to the quantum numbers of the hadrons. The quark model underlies "flavor SU(3)", or the E ...

. In particular, the

''Q'' of a quark or hadron particle is related to its

''I₃'' and its

hypercharge In particle physics, the hypercharge (a portmanteau of hyperonic and charge) ''Y'' of a particle is a quantum number conserved under the strong interaction. The concept of hypercharge provides a single charge operator that accounts for propert ...

''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 Charge or charged may refer to: Arts, entertainment, and media Films * ''Charge, Zero Emissions/Maximum Speed'', a 2011 documentary Music * ''Charge'' (David Ford album) * ''Charge'' (Machel Montano album) * ''Charge!!'', an album by The Aqua ...

, ''I''₃ the 3rd-component of the

, ''B'' the

, and ''S'', ''C'', ''B′'', ''T'' are the

and

numbers. Expressed in terms of quark content, these would become: :

\\ 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 theory 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 ...

point of view, the Gell-Mann–Nishijima formula and its generalized version can be derived using an approximate

SU(3) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special ...

flavour symmetry 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 numbers'' th ...

because the charges can be defined using the corresponding conserved

Noether current Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...

Formula

References

Further reading