Editing Nucleon (section)

==Models==<!-- This section is linked from [[Casimir effect]] -->
{{confusing|section|date=August 2007}}

Although it is known that the nucleon is made from three quarks, {{As of|2006|lc=on}}, it is not known how to solve the [[equations of motion]] for [[quantum chromodynamics]]. Thus, the study of the low-energy properties of the nucleon are performed by means of models. The only first-principles approach available is to attempt to solve the equations of QCD numerically, using [[lattice QCD]]. This requires complicated algorithms and very powerful [[supercomputer]]s. However, several analytic models also exist:

===Skyrmion models===
The [[skyrmion]] models the nucleon as a [[topological soliton]] in a nonlinear [[SU(2)]] [[pion]] field. The topological stability of the skyrmion is interpreted as the conservation of [[baryon number]], that is, the non-decay of the nucleon. The local [[topological winding number]] density is identified with the local [[baryon number]] density of the nucleon. With the pion isospin vector field oriented in the shape of a [[hedgehog space]], the model is readily solvable, and is thus sometimes called the ''hedgehog model''. The hedgehog model is able to predict low-energy parameters, such as the nucleon mass, radius and [[axial coupling constant]], to approximately 30% of experimental values.

===MIT bag model===
The ''MIT bag model''<ref>Chodos et al. [https://doi.org/10.1103/PhysRevD.9.3471 "New extended model of hadrons"] {{Webarchive|url=https://web.archive.org/web/20231230134936/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.9.3471 |date=2023-12-30 }}, Phys. Rev. D 9, 3471 (1974).</ref><ref>Chodos et al. [https://doi.org/10.1103/PhysRevD.10.2599 "Baryon structure in the bag theory"] {{Webarchive|url=https://web.archive.org/web/20231230134924/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.10.2599 |date=2023-12-30 }}, Phys. Rev. D 10, 2599 (1974).</ref><ref>DeGrand et al. [https://doi.org/10.1103/PhysRevD.12.2060 "Masses and other parameters of the light hadrons"] {{Webarchive|url=https://web.archive.org/web/20231230134817/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.12.2060 |date=2023-12-30 }}, Phys. Rev. D 12, 2060 (1975).</ref> confines quarks and gluons interacting through [[quantum chromodynamics]] to a region of space determined by balancing the pressure exerted by the quarks and gluons against a hypothetical pressure exerted by the vacuum on all colored quantum fields.  The simplest approximation to the model confines three non-interacting quarks to a spherical cavity, with the [[boundary condition]] that the quark [[vector current]] vanish on the boundary. The non-interacting treatment of the quarks is justified by appealing to the idea of [[asymptotic freedom]], whereas the hard-boundary condition is justified by [[quark confinement]].

Mathematically, the model vaguely resembles that of a [[radar cavity]], with solutions to the [[Dirac equation]] standing in for solutions to the [[Maxwell equations]], and the vanishing vector current boundary condition standing for the conducting metal walls of the radar cavity. If the radius of the bag is set to the radius of the nucleon, the '''bag model''' predicts a nucleon mass that is within 30% of the actual mass.

Although the basic bag model does not provide a pion-mediated interaction, it describes excellently the nucleon–nucleon forces through the 6&nbsp;quark bag ''s''-channel mechanism using the ''P''-matrix.<ref>{{cite journal |last1=Jaffe |first1=R. L. |author1-link=Robert Jaffe (physicist) |author2-link=Francis E. Low |last2=Low |first2=F. E. |year=1979 |title=Connection between quark-model eigenstates and low-energy scattering |journal=Phys. Rev. D |volume=19 |issue=7| page=2105 |doi=10.1103/PhysRevD.19.2105 |bibcode=1979PhRvD..19.2105J }}</ref><ref>
{{cite journal |last1=Yu |last2=Simonov |first2=A. |year=1981 |title=The quark compound bag model and the Jaffe-Low ''P''-matrix |journal=[[Physics Letters B]] |volume=107 |issue=1–2 |page=1 |doi=10.1016/0370-2693(81)91133-3 |bibcode=1981PhLB..107....1S}}</ref>

===Chiral bag model===
The ''chiral bag model''<ref>{{cite journal
 |author1-link=Gerald E. Brown  |first1=Gerald E. |last1=Brown
 |author2-link=Mannque Rho  |first2=Mannque |last2=Rho
 |date=March 1979
 |title=The little bag
 |journal=[[Physics Letters B]]
 |volume=82  |issue=2  |pages=177–180
 |doi=10.1016/0370-2693(79)90729-9
 |bibcode=1979PhLB...82..177B
}}</ref><ref>{{cite journal
 |last1=Vepstas |first1=L.
 |last2=Jackson |first2=A. D.
 |last3=Goldhaber |first3=A. S.
 |year=1984
 |title=Two-phase models of baryons and the chiral Casimir effect
 |journal=[[Physics Letters B]]
 |volume=140  |issue=5–6  |pages=280–284
 |bibcode=1984PhLB..140..280V
 |doi=10.1016/0370-2693(84)90753-6
}}</ref> merges the ''MIT bag model'' and the ''skyrmion model''. In this model, a hole is punched out of the middle of the skyrmion and replaced with a bag model. The boundary condition is provided by the requirement of continuity of the [[axial vector current]] across the bag boundary.

Very curiously, the missing part of the topological winding number (the baryon number) of the hole punched into the skyrmion is exactly made up by the non-zero [[vacuum expectation value]] (or [[spectral asymmetry]]) of the quark fields inside the bag. {{As of|2017}}, this remarkable trade-off between [[topology]] and the [[spectrum of an operator]] does not have any grounding or explanation in the mathematical theory of [[Hilbert space]]s and their relationship to [[geometry]].

Several other properties of the chiral bag are notable: It provides a better fit to the low-energy nucleon properties, to within 5–10%, and these are almost completely independent of the chiral-bag radius, as long as the radius is less than the nucleon radius. This independence of radius is referred to as the ''Cheshire Cat principle'',<ref>{{cite journal
 |last1=Vepstas |first1=L.
 |last2=Jackson |first2=A. D.
 |year=1990
 |title=Justifying the chiral bag
 |journal=[[Physics Reports]]
 |volume=187  |issue=3  |pages=109–143
 |bibcode=1990PhR...187..109V
 |doi=10.1016/0370-1573(90)90056-8
}}</ref> after the fading of [[Lewis Carroll]]'s [[Cheshire Cat]] to just its smile. It is expected that a first-principles solution of the equations of QCD will demonstrate a similar duality of quark–[[meson]] descriptions.