# Cornell potential

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{{Short description|Simple potential between quarks}}
In [particle physics](/source/particle_physics), the '''Cornell potential''' is an effective method to account for the [confinement](/source/Color_confinement) of [quarks](/source/quarks) in [quantum chromodynamics](/source/quantum_chromodynamics) (QCD). It was developed by  [Estia J. Eichten](/source/Estia_J._Eichten), [Kurt Gottfried](/source/Kurt_Gottfried),  [Toichiro Kinoshita](/source/Toichiro_Kinoshita), [John Kogut](/source/John_Kogut), [Kenneth Lane](/source/Kenneth_Lane_(physicist)) and [Tung-Mow Yan](/source/Tung-Mow_Yan) at [Cornell University](/source/Cornell_University)<ref>
{{cite journal
 |last1=Eichten |first1=E.
 |last2=Gottfried |first2=K.
 |last3=Kinoshita |first3=T.
 |last4=Kogut |first4=J. B.
 |last5=Lane |first5=K. D.
 |last6=Yan |first6=T. M.
 |title=Spectrum of charmed quark-antiquark bound states
 |journal=Phys. Rev. Lett. 
 |volume=34 |issue=369
 |year=1975
 |page=369
 |doi=10.1103/PhysRevLett.34.369
 |bibcode=1975PhRvL..34..369E
 }}</ref><ref>
{{cite journal
 |last1=Eichten |first1=E.
 |last2=Gottfried |first2=K.
 |last3=Kinoshita |first3=T.
 |last4=Lane |first4=K. D.
 |last5=Yan |first5=T. M.
 |title=Charmonium: The model
 |journal=Phys. Rev. D
 |volume=17 |issue=3090
 |year=1978
 |page=3090
 |doi=10.1103/PhysRevD.17.3090
 |bibcode=1978PhRvD..17.3090E
 }}</ref> in the 1970s to explain the masses of [quarkonium](/source/quarkonium) states and account for the relation between the mass and [angular momentum](/source/angular_momentum) of the [hadron](/source/hadron) (the so-called [Regge trajectories](/source/Regge_theory)). The potential has the form:<ref name="HUGS98">
{{cite journal
 |last1=Brambilla |first1=N. 
 |last2=Vairo |first2=A.
 |title=Quark confinement and the hadron spectrum
 |journal=Proceedings of the 13th Annual HUGS AT CEBAF
 |year=1998
 |arxiv=hep-ph/9904330
 |doi=
 }}</ref>

:<math>V(r) = -\frac{4}{3}\frac{\alpha_s}{\;r\;} + \sigma\,r + \text{constant}</math>

where <math>r</math> is the effective radius of the quarkonium state, <math>\alpha_s</math> is the QCD [running coupling](/source/running_coupling), <math>\sigma</math> is the [QCD string](/source/QCD_string) tension and is a constant of <math>\simeq 0.18 GeV^2</math>.<ref name="alpha_s review 2016"/> Initially, <math>\alpha_s</math> and <math>\sigma</math> were merely empirical parameters but with the development of QCD can now be calculated using [perturbative QCD](/source/Perturbative_quantum_chromodynamics) and [lattice QCD](/source/lattice_QCD), respectively.

==Short distance potential==

The potential consists of two parts. The first one, <math>-\frac{4}{3}\frac{\alpha_s}{\;r\;}</math> dominate at short distances, typically for <math>r <0.1</math> fm.<ref name="HUGS98" /> It arises from the one-[gluon](/source/gluon) exchange between the quark and its anti-quark, and is known as the Coulombic part of the potential, since it has the same form as the well-known [Coulombic potential](/source/Coulomb's_law) <math>\;\frac{\alpha}{\;r\;}\;</math> induced by the [electromagnetic force](/source/Electrostatics) (where <math>\alpha</math> is the [electromagnetic coupling constant](/source/Fine-structure_constant)).

The factor <math>\frac{4}{3}</math> in QCD comes from the fact that quarks have different type of [charges](/source/Charge_(physics)) (''[colors](/source/Color_charge)'') and is associated with any [gluon](/source/gluon) emission from a quark. Specifically, this factor is called the ''color factor'' or ''[Casimir](/source/Hendrik_Casimir) factor'' and is <math>  C_F \equiv  \frac{N_c^2-1}{2N_c}= \frac{4}{3}</math>, where <math>N_c = 3</math> is the number of color charges.

The value for <math>\alpha_s</math> depends on the radius of the studied hadron. Its value ranges from 0.19 to 0.4.<ref name="alpha_s review 2016">
{{cite journal
 |last1=Deur |first1=A.
 |last2=Brodsky |first2=S. J.
 |last3=de Teramond |first3=G. F.
 |title=The QCD Running Coupling
 |journal=Prog. Part. Nucl. Phys.
 |volume=90 |issue=1
 |year=2016
 |pages=1–74
 |arxiv=1604.08082
 |doi=10.1016/j.ppnp.2016.04.003
 |bibcode=2016PrPNP..90....1D
 |s2cid=118854278
 }}</ref> 
For precise determination of the short distance potential, the [running](/source/Running_coupling) of <math>\alpha_s</math> must be accounted for, resulting in a distant-dependent <math>\alpha_s(r)</math>. Specifically, <math>\alpha_s</math> must be calculated in the so-called ''potential [renormalization scheme](/source/renormalization)'' (also denoted V-scheme) and, since [quantum field theory](/source/quantum_field_theory) calculations are usually done in [momentum space](/source/momentum_space), [Fourier transform](/source/Fourier_transform)ed to [position space](/source/momentum_space).<ref name="alpha_s review 2016"/>

==Long distance potential==

The second term of the potential, <math>\sigma\,r</math>, is the linear confinement term and fold-in the non-[perturbative QCD](/source/Perturbative_quantum_chromodynamics) effects that result in color confinement. <math>\sigma</math> is interpreted as the tension of the [QCD string](/source/QCD_string) that forms when the gluonic [field line](/source/field_line)s collapse into a [flux tube](/source/flux_tube). Its value is <math> \sigma \sim 0.18</math> GeV<math>^2</math>.<ref name="alpha_s review 2016"/> 
<math>\sigma</math> controls the intercepts and slopes of the linear [Regge trajectories](/source/Regge_theory).

==Domains of application==

The Cornell potential applies best for the case of static quarks (or very heavy quarks with non-[relativistic](/source/Special_relativity) motion), although relativistic improvements to the potential using speed-dependent terms are available.<ref name="HUGS98" /> Likewise, the potential has been extended to include [spin](/source/Spin_(physics))-dependent terms<ref name="HUGS98" />

==Calculation of the quark-quark potential==
A test of validity for approaches that seek to explain [color confinement](/source/color_confinement) is that they must produce, in the limit that quark motions are non-relativistic, a potential that agrees with the Cornell potential.

A significant achievement of [lattice QCD](/source/lattice_QCD) is to be able compute from first principles the static quark-antiquark potential, with results confirming the empirical Cornell Potential.<ref>
{{cite journal
 |last1=Bali |first1=G. S. 
 |title=QCD forces and heavy quark bound states
 |journal=Phys. Rep. 
 |volume=343 |issue=1
 |year=2001
 |pages=1–136 
 |arxiv=hep-ph/0001312
 |doi=10.1016/S0370-1573(00)00079-X
 |bibcode=2001PhR...343....1B 
 |s2cid=119050904 
 }}</ref>

Other approaches to the confinement problem also results in the Cornell potential, including the [dual superconductor model](/source/dual_superconductor_model), the [Abelian Higgs model](/source/Abelian_Higgs_model), and the [center vortex models](/source/QCD_vacuum).<ref name="HUGS98" /><ref>{{Cite book<!--Deny Citation Bot-->
 |last=Greensite |first=J.
 |year=2011
 |title=An introduction to the confinement problem
 |series=Lecture Notes in Physics
 |volume=821
 |publisher=[Springer](/source/Springer_Science%2BBusiness_Media)
 |isbn=978-3-642-14381-6
 |bibcode=2011LNP...821.....G
 |doi=10.1007/978-3-642-14382-3
}}</ref>

More recently, calculations based on the [AdS/CFT correspondence](/source/AdS%2FCFT_correspondence) have reproduced the Cornell potential using the [AdS/QCD correspondence](/source/AdS%2FQCD_correspondence)<ref>{{ cite journal | doi=10.1103/PhysRevD.74.015005 |author1=A. Karch |author2=E. Katz |author3=D. T. Son |author4=M. A. Stephanov | title=Linear Confinement and AdS/QCD |journal=[Physical Review D](/source/Physical_Review_D) | volume=74 | date=2006 |issue=1 | article-number=015005 | arxiv=hep-ph/0602229|bibcode = 2006PhRvD..74a5005K |s2cid=16228097 }}</ref><ref>
{{cite journal
 |last1=Andreev |first1=O.
 |last2=Zakharov |first2=V. I.
 |title=Heavy-quark potentials and AdS/QCD
 |journal=Phys. Rev. D
 |volume=74 |issue=25023
 |year=2006
 |article-number=025023
 |arxiv=hep-ph/0604204
 |doi=10.1103/PhysRevD.74.025023
 |bibcode=2006PhRvD..74b5023A
 |s2cid=119391222
 }}</ref>
or [light front holography](/source/light_front_holography).<ref>
{{cite journal
 |last1=Trawinski |first1=A. P.
 |last2=Glazek |first2=S. D.
 |last3=Brodsky |first3=S. J.
 |last4=de Teramond |first4=G. F.
 |last5=Dosch |first5=H. G.
 |title=Effective confining potentials for QCD
 |journal=Phys. Rev. D
 |volume=90 |issue=74017
 |year=2014
 |article-number=074017
 |arxiv=1403.5651
 |doi=10.1103/PhysRevD.90.074017
 |bibcode=2014PhRvD..90g4017T
 |s2cid=118644867
 }}</ref>

==See also==
*[Color confinement](/source/Color_confinement)
*[QCD vacuum](/source/QCD_vacuum)

== References ==
{{reflist}}

Category:Quantum chromodynamics
Category:Hadrons
Category:Mesons
Category:Quantum mechanical potentials

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Adapted from the Wikipedia article [Cornell potential](https://en.wikipedia.org/wiki/Cornell_potential) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Cornell_potential?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.