Editing Baryon (section)

== Properties ==

=== Isospin and charge ===
{{Main|Isospin}}

[[File:Baryon-decuplet-small.svg|thumb|200px|
Combinations of three '''[[up quark|u]], [[down quark|d]]''' or '''[[strange quark|s]]''' quarks forming baryons with a spin-{{sfrac|3|2}} form the ''[[Eightfold way (physics)|uds baryon decuplet]]'']]
[[File:Baryon-octet-small.svg|thumb|200px|Combinations of three '''[[up quark|u]], [[down quark|d]]''' or '''[[strange quark|s]]''' quarks forming baryons with a spin-{{sfrac|1|2}} form the ''[[Eightfold way (physics)|uds baryon octet]]'']]

The concept of isospin was first proposed by [[Werner Heisenberg]] in 1932 to explain the similarities between protons and neutrons under the [[strong interaction]].<ref>W. Heisenberg (1932)</ref> Although they had different electric charges, their masses were so similar that physicists believed they were the same particle. The different electric charges were explained as being the result of some unknown excitation similar to spin. This unknown excitation was later dubbed ''isospin'' by [[Eugene Wigner]] in 1937.<ref>E. Wigner (1937)</ref>

This belief lasted until [[Murray Gell-Mann]] proposed the [[quark model]] in 1964 (containing originally only the u, d, and s quarks).<ref>M. Gell-Mann (1964)</ref> The success of the isospin model is now understood to be the result of the similar masses of u and d quarks. Since u and d quarks have similar masses, particles made of the same number then also have similar masses. The exact specific u and d quark composition determines the charge, as u quarks carry charge +{{sfrac|2|3}} while d quarks carry charge −{{sfrac|1|3}}. For example, the four [[Delta baryon|Deltas]] all have different charges ({{SubatomicParticle|Delta++}} (uuu), {{SubatomicParticle|Delta+}} (uud), {{SubatomicParticle|Delta0}} (udd), {{SubatomicParticle|Delta-}} (ddd)), but have similar masses (~1,232&nbsp;MeV/c<sup>2</sup>) as they are each made of a combination of three u or d quarks. Under the isospin model, they were considered to be a single particle in different charged states.

The mathematics of isospin was modeled after that of spin. Isospin projections varied in increments of 1 just like those of spin, and to each projection was associated a "[[Quantum state|charged state]]". Since the "[[Delta baryon|Delta particle]]" had four "charged states", it was said to be of isospin ''I''&nbsp;=&nbsp;{{sfrac|3|2}}. Its "charged states" {{SubatomicParticle|Delta++}}, {{SubatomicParticle|Delta+}}, {{SubatomicParticle|Delta0}}, and {{SubatomicParticle|Delta-}}, corresponded to the isospin projections ''I''<sub>3</sub>&nbsp;=&nbsp;+{{sfrac|3|2}}, ''I''<sub>3</sub>&nbsp;=&nbsp;+{{sfrac|1|2}}, ''I''<sub>3</sub>&nbsp;=&nbsp;−{{sfrac|1|2}}, and ''I''<sub>3</sub>&nbsp;=&nbsp;−{{sfrac|3|2}}, respectively. Another example is the "nucleon particle". As there were two nucleon "charged states", it was said to be of isospin {{sfrac|1|2}}. The positive nucleon {{SubatomicParticle|Nucleon+}} (proton) was identified with ''I''<sub>3</sub>&nbsp;=&nbsp;+{{sfrac|1|2}} and the neutral nucleon {{SubatomicParticle|Nucleon0}} (neutron) with ''I''<sub>3</sub>&nbsp;=&nbsp;−{{sfrac|1|2}}.<ref name=WongA>S.S.M. Wong (1998a)</ref> It was later noted that the isospin projections were related to the up and down quark content of particles by the relation:
: <math>I_\mathrm{3}=\frac{1}{2}[(n_\mathrm{u}-n_\mathrm{\bar{u}})-(n_\mathrm{d}-n_\mathrm{\bar{d}})],</math>
where the ''n'''s are the number of up and down quarks and antiquarks.

In the "isospin picture", the four Deltas and the two nucleons were thought to be the different states of two particles. However, in the quark model, Deltas are different states of nucleons (the N<sup>++</sup> or N<sup>−</sup> are forbidden by [[Pauli's exclusion principle]]). Isospin, although conveying an inaccurate picture of things, is still used to classify baryons, leading to unnatural and often confusing nomenclature.

=== Flavour quantum numbers ===
{{Main|Flavour (particle physics)#Flavour quantum numbers}}

The [[strangeness]] [[flavour (particle physics)#Flavour quantum numbers|flavour quantum number]] ''S'' (not to be confused with spin) was noticed to go up and down along with particle mass. The higher the mass, the lower the strangeness (the more s quarks). Particles could be described with isospin projections (related to charge) and strangeness (mass) (see the uds [[Eightfold way (physics)#Baryon octet|octet]] and [[Eightfold way (physics)#Baryon decuplet|decuplet]] figures on the right). As other quarks were discovered, new quantum numbers were made to have similar description of udc and udb octets and decuplets. Since only the u and d mass are similar, this description of particle mass and charge in terms of isospin and flavour quantum numbers works well only for octet and decuplet made of one u, one d, and one other quark, and breaks down for the other octets and decuplets (for example, ucb octet and decuplet). If the quarks all had the same mass, their behaviour would be called ''symmetric'', as they would all behave in the same way to the strong interaction. Since quarks do not have the same mass, they do not interact in the same way (exactly like an electron placed in an electric field will accelerate more than a proton placed in the same field because of its lighter mass), and the symmetry is said to be [[broken symmetry|broken]].

It was noted that charge (''Q'') was related to the isospin projection (''I''<sub>3</sub>), the [[baryon number]] (''B'') and flavour quantum numbers (''S'', ''C'', ''B''′, ''T'') by the [[Gell-Mann–Nishijima formula]]:<ref name="WongA" />
: <math>Q = I_3 +\frac{1}{2}\left(B + S + C + B^\prime + T\right),</math>
where ''S'', ''C'', ''B''′, and ''T'' represent the [[strangeness]], [[charm (quantum number)|charm]], [[bottomness]] and [[topness]] flavour quantum numbers, respectively. They are related to the number of strange, charm, bottom, and top quarks and antiquark according to the relations:
: <math>\begin{align}
         S &= -\left(n_\mathrm{s} - n_\mathrm{\bar{s}}\right), \\
         C &= +\left(n_\mathrm{c} - n_\mathrm{\bar{c}}\right), \\
  B^\prime &= -\left(n_\mathrm{b} - n_\mathrm{\bar{b}}\right), \\
         T &= +\left(n_\mathrm{t} - n_\mathrm{\bar{t}}\right),
\end{align}</math>
meaning that the Gell-Mann–Nishijima formula is equivalent to the expression of charge in terms of quark content:
: <math>Q = \frac{2}{3}\left[(n_\mathrm{u} - n_\mathrm{\bar{u}}) + (n_\mathrm{c} - n_\mathrm{\bar{c}}) + (n_\mathrm{t} - n_\mathrm{\bar{t}})\right] - \frac{1}{3}\left[(n_\mathrm{d} - n_\mathrm{\bar{d}}) + (n_\mathrm{s} - n_\mathrm{\bar{s}}) + (n_\mathrm{b} - n_\mathrm{\bar{b}})\right].</math>

=== Spin, orbital angular momentum, and total angular momentum ===
{{Main|Spin (physics)|Angular momentum operator|Quantum numbers|Clebsch–Gordan coefficients}}
[[Spin (physics)|Spin]] (quantum number ''S'') is a [[Euclidean vector|vector]] quantity that represents the "intrinsic" [[angular momentum]] of a particle. It comes in increments of {{sfrac|1|2}}&nbsp;[[Planck constant|ħ]] (pronounced "h-bar"). The ħ is often dropped because it is the "fundamental" unit of spin, and it is implied that "spin 1" means "spin 1&nbsp;ħ". In some systems of [[natural units]], ħ is chosen to be 1, and therefore does not appear anywhere.

[[Quark]]s are [[fermion]]ic particles of spin {{sfrac|1|2}} (''S''&nbsp;=&nbsp;{{sfrac|1|2}}). Because spin projections vary in increments of 1 (that is 1&nbsp;ħ), a single quark has a spin vector of length {{sfrac|1|2}}, and has two spin projections (''S''<sub>z</sub>&nbsp;=&nbsp;+{{sfrac|1|2}} and ''S''<sub>z</sub>&nbsp;=&nbsp;−{{sfrac|1|2}}). Two quarks can have their spins aligned, in which case the two spin vectors add to make a vector of length ''S''&nbsp;=&nbsp;1 and three spin projections (''S''<sub>z</sub>&nbsp;=&nbsp;+1, ''S''<sub>z</sub>&nbsp;=&nbsp;0, and ''S''<sub>z</sub>&nbsp;=&nbsp;−1). If two quarks have unaligned spins, the spin vectors add up to make a vector of length ''S''&nbsp;=&nbsp;0 and has only one spin projection (''S''<sub>z</sub>&nbsp;=&nbsp;0), etc. Since baryons are made of three quarks, their spin vectors can add to make a vector of length ''S''&nbsp;=&nbsp;{{sfrac|3|2}}, which has four spin projections (''S''<sub>z</sub>&nbsp;=&nbsp;+{{sfrac|3|2}}, ''S''<sub>z</sub>&nbsp;=&nbsp;+{{sfrac|1|2}}, ''S''<sub>z</sub>&nbsp;=&nbsp;−{{sfrac|1|2}}, and ''S''<sub>z</sub>&nbsp;=&nbsp;−{{sfrac|3|2}}), or a vector of length ''S''&nbsp;=&nbsp;{{sfrac|1|2}} with two spin projections (''S''<sub>z</sub>&nbsp;=&nbsp;+{{sfrac|1|2}}, and ''S''<sub>z</sub>&nbsp;=&nbsp;−{{sfrac|1|2}}).<ref name=Shankar>R. Shankar (1994)</ref>

There is another quantity of angular momentum, called the [[angular momentum operator|orbital angular momentum]] ([[azimuthal quantum number]] ''L''), that comes in increments of 1&nbsp;ħ, which represent the angular moment due to quarks orbiting around each other. The [[angular momentum operator|total angular momentum]] ([[total angular momentum quantum number]] ''J'') of a particle is therefore the combination of intrinsic angular momentum (spin) and orbital angular momentum. It can take any value from {{nowrap|''J'' {{=}} {{!}}''L'' − ''S''{{!}}}} to {{nowrap|''J'' {{=}} {{!}}''L'' + ''S''{{!}}}}, in increments of 1.
{|class="wikitable" style="margin:1em auto; text-align: center;"
|+ Baryon angular momentum quantum numbers for ''L'' = 0, 1, 2, 3
|-
! Spin, <br />''S''
! Orbital angular <br />momentum, ''L''
! Total angular <br />momentum, ''J''
! [[#Parity|Parity]], <br />''P''
! Condensed <br />notation, ''J''<sup>''P''</sup>
|-
|rowspan="4"| {{sfrac|1|2}}
| 0 || {{sfrac|1|2}} || + || {{sfrac|1|2}}<sup>+</sup>
|-
| 1 || {{sfrac|3|2}}, {{sfrac|1|2}} || − || {{sfrac|3|2}}<sup>−</sup>, {{sfrac|1|2}}<sup>−</sup>
|-
| 2 || {{sfrac|5|2}}, {{sfrac|3|2}} || + || {{sfrac|5|2}}<sup>+</sup>, {{sfrac|3|2}}<sup>+</sup>
|-
| 3 || {{sfrac|7|2}}, {{sfrac|5|2}} || − || {{sfrac|7|2}}<sup>−</sup>, {{sfrac|5|2}}<sup>−</sup>
|-
|rowspan="4"| {{sfrac|3|2}}
| 0 || {{sfrac|3|2}} || + || {{sfrac|3|2}}<sup>+</sup>
|-
| 1 || {{sfrac|5|2}}, {{sfrac|3|2}}, {{sfrac|1|2}} || − || {{sfrac|5|2}}<sup>−</sup>, {{sfrac|3|2}}<sup>−</sup>, {{sfrac|1|2}}<sup>−</sup>
|-
| 2 || {{sfrac|7|2}}, {{sfrac|5|2}}, {{sfrac|3|2}}, {{sfrac|1|2}} || + || {{sfrac|7|2}}<sup>+</sup>, {{sfrac|5|2}}<sup>+</sup>, {{sfrac|3|2}}<sup>+</sup>, {{sfrac|1|2}}<sup>+</sup>
|-
| 3 || {{sfrac|9|2}}, {{sfrac|7|2}}, {{sfrac|5|2}}, {{sfrac|3|2}} || − || {{sfrac|9|2}}<sup>−</sup>, {{sfrac|7|2}}<sup>−</sup>, {{sfrac|5|2}}<sup>−</sup>, {{sfrac|3|2}}<sup>−</sup>
|}

Particle physicists are most interested in baryons with no orbital angular momentum (''L''&nbsp;=&nbsp;0), as they correspond to [[ground state]]s—states of minimal energy. Therefore, the two groups of baryons most studied are the ''S''&nbsp;=&nbsp;{{sfrac|1|2}}; ''L''&nbsp;=&nbsp;0 and ''S''&nbsp;=&nbsp;{{sfrac|3|2}}; ''L''&nbsp;=&nbsp;0, which corresponds to ''J''&nbsp;=&nbsp;{{sfrac|1|2}}<sup>+</sup> and ''J''&nbsp;=&nbsp;{{sfrac|3|2}}<sup>+</sup>, respectively, although they are not the only ones. It is also possible to obtain ''J''&nbsp;=&nbsp;{{sfrac|3|2}}<sup>+</sup> particles from ''S''&nbsp;=&nbsp;{{sfrac|1|2}} and ''L''&nbsp;=&nbsp;2, as well as ''S''&nbsp;=&nbsp;{{sfrac|3|2}} and ''L''&nbsp;=&nbsp;2. This phenomenon of having multiple particles in the same total angular momentum configuration is called ''[[degenerate energy level|degeneracy]]''. How to distinguish between these degenerate baryons is an active area of research in [[baryon spectroscopy]].<ref>H. Garcilazo et al. (2007)</ref><ref>D.M. Manley (2005)</ref>

=== Parity ===
{{Main|Parity (physics)}}

If the universe were reflected in a mirror, most of the laws of physics would be identical—things would behave the same way regardless of what we call "left" and what we call "right". This concept of mirror reflection is called "[[parity (physics)|intrinsic parity]]" or simply "parity" (''P''). [[Gravity]], the [[electromagnetic force]], and the [[strong interaction]] all behave in the same way regardless of whether or not the universe is reflected in a mirror, and thus are said to [[P-symmetry|conserve parity]] (P-symmetry). However, the [[weak interaction]] does distinguish "left" from "right", a phenomenon called [[parity violation]] (P-violation).

Based on this, if the [[wavefunction]] for each particle (in more precise terms, the [[quantum field]] for each particle type) were simultaneously mirror-reversed, then the new set of wavefunctions would perfectly satisfy the laws of physics (apart from the weak interaction). It turns out that this is not quite true: for the equations to be satisfied, the wavefunctions of certain types of particles have to be multiplied by −1, in addition to being mirror-reversed. Such particle types are said to have negative or odd parity (''P''&nbsp;=&nbsp;−1, or alternatively ''P''&nbsp;=&nbsp;–), while the other particles are said to have positive or even parity (''P''&nbsp;=&nbsp;+1, or alternatively ''P''&nbsp;=&nbsp;+).

For baryons, the parity is related to the orbital angular momentum by the relation:<ref name=WongB>S.S.M. Wong (1998b)</ref>
: <math>P=(-1)^L.\ </math>

As a consequence, baryons with no orbital angular momentum (''L''&nbsp;=&nbsp;0) all have even parity (''P''&nbsp;=&nbsp;+).