Gell-Mann–Okubo Mass Formula
   HOME

TheInfoList



OR:

In
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
, the Gell-Mann–Okubo mass formula provides a sum rule for the masses of
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 within a specific
multiplet In physics and particularly in particle physics, a multiplet is the state space for 'internal' degrees of freedom of a particle; that is, degrees of freedom associated to a particle itself, as opposed to 'external' degrees of freedom such as th ...
, determined by their
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 ...
(''I'') and
strangeness In particle physics, strangeness (symbol ''S'') is a property of particles, expressed as a quantum number, for describing decay of particles in strong and electromagnetic interactions that occur in a short period of time. The strangeness of a ...
(or alternatively,
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 ...
) :M = a_0 + a_1 Y + a_2 \left I \left( I + 1 \right)-\frac Y^2 \right, where ''a''0, ''a''1, and ''a''2 are
free parameter A free parameter is a variable in a mathematical model which cannot be predicted precisely or constrained by the model and must be estimated experimentally or theoretically. A mathematical model, theory, or conjecture is more likely to be right a ...
s. The rule was first formulated by
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 ...
in 1961 and independently proposed by Susumu Okubo in 1962. Isospin and hypercharge are generated by
SU(3) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 i ...
, which can be represented by eight hermitian and traceless matrices corresponding to the "components" of isospin and hypercharge. Six of the matrices correspond to flavor change, and the final two correspond to the third-component of isospin projection, and hypercharge.


Theory

The mass formula was obtained by considering the
representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...
of the
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
su(3) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 i ...
. In particular, the
meson In particle physics, a meson () is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticles, the ...
octet corresponds to the
root system In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and ...
of the
adjoint representation In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is \m ...
. However, the simplest, lowest-dimensional representation of su(3) is the
fundamental representation In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight. For example, the defi ...
, which is three-dimensional, and is now understood to describe the approximate
flavor 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'' ...
of the three
quark 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 nucleus, atomic nuclei ...
s ''u'', ''d'', and ''s''. Thus, the discovery of not only an su(3) symmetry, but also of this workable formula for the
mass spectrum A mass spectrum is a histogram plot of intensity vs. ''mass-to-charge ratio'' (''m/z'') in a chemical sample, usually acquired using an instrument called a ''mass spectrometer''. Not all mass spectra of a given substance are the same; for example ...
was one of the earliest indicators for the existence of quarks. The formula is underlain by the ''octet enhancement hypothesis'', which ascribes dominance of SU(3) breaking to the hypercharge generator of SU(3), Y=\tfracF_8=\operatorname(1,1,-2)/3~, and, in modern terms, the relatively higher mass of the
strange quark The strange quark or s quark (from its symbol, s) is the third lightest of all quarks, a type of elementary particle. Strange quarks are found in subatomic particles called hadrons. Examples of hadrons containing strange quarks include kaons (), ...
. This formula is ''phenomenological'', describing an approximate relation between meson and baryon masses, and has been superseded as theoretical work in
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 ...
advances, notably chiral perturbation theory.


Baryons

Using the values of relevant ''I'' and ''S'' for baryons, the Gell-Mann–Okubo formula can be rewritten for the baryon octet, :\frac = \frac \, where ''N'', Λ, Σ, and Ξ represent the average mass of corresponding baryons. Using the current mass of baryons, this yields: : \frac = 1128.5~\mathrm/c^2 and : \frac = 1135.25~\mathrm/c^2 meaning that the Gell-Mann–Okubo formula reproduces the mass of octet baryons within ~0.5% of measured values. For the baryon decuplet, the Gell-Mann–Okubo formula can be rewritten as the "equal-spacing" rule :\Delta -\Sigma^* = \Sigma^* - \Xi^* = \Xi^* - \Omega = a_1 + 2a_2 \approx \, -147 ~\mathrm/c^2 where Δ, Σ*, Ξ*, and Ω represent the average mass of corresponding baryons. The baryon decuplet formula famously allowed Gell-Mann to predict the mass of the then undiscovered Ω.


Mesons

The same mass relation can be found for the meson octet, :\frac\left(\frac + \frac\right) = \frac Using the current mass of mesons, this yields : \frac\left(\frac + \frac\right)= 496~\mathrm/c^2 and : \frac = 445~\mathrm/c^2 Because of this large discrepancy, several people attempted to find a way to understand the failure of the GMO formula in mesons, when it worked so well in baryons. In particular, people noticed that using the square of the average masses yielded much better results: :\frac\left \left( \frac \right)^2 + \left( \frac \right)^2\right= \frac This now yields :\frac\left \left( \frac \right)^2 + \left( \frac \right)^2\right 246\times10^3~\mathrm/c^4 and :\frac = 230\times10^3~\mathrm/c^4 which fall within 5% of each other. For a while, the GMO formula involving the square of masses was simply an
empirical relationship In science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural science ...
; but later a justification for using the square of masses was found in the context of
chiral perturbation theory Chiral perturbation theory (ChPT) is an effective field theory constructed with a Lagrangian (field theory), Lagrangian consistent with the (approximate) chiral symmetry of quantum chromodynamics (QCD), as well as the other symmetries of parity (ph ...
, just for pseudoscalar mesons, since these are the pseudogoldstone bosons of dynamically broken
chiral symmetry A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particl ...
, and, as such, obey Dashen's mass formula. Other, mesons, such as vector ones, need no squaring for the GMO formula to work.


See also

*
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 quarks and hadrons to the electric charge ''Q''. It was originally given by Kazuhiko Nishijima a ...
* Eightfold Way *
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 ...
*
SU(3) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The matrices of the more general unitary group may have complex determinants with absolute value 1, rather than real 1 i ...


References


Further reading

The following book contains most (if not all) historical papers on the Eightfold Way and related topics, including the Gell-Mann–Okubo mass formula. * {{DEFAULTSORT:Gell-Mann-Okubo mass formula Hadrons Quantum chromodynamics Murray Gell-Mann