Cornell potential

The potential consists of two parts. The first one, ${\displaystyle -{\frac {4}{3}}{\frac {\alpha _{s}}{\;r\;}}}$  dominate at short distances, typically for ${\displaystyle r<0.1}$  fm.^[3] It arises from the one-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 ${\displaystyle \;{\frac {\alpha }{\;r\;}}\;}$  induced by the electromagnetic force (where ${\displaystyle \alpha }$  is the electromagnetic coupling constant).

The factor ${\displaystyle {\frac {4}{3}}}$  in QCD comes from the fact that quarks have different type of charges (colors) and is associated with any gluon emission from a quark. Specifically, this factor is called the color factor or Casimir factor and is ${\displaystyle C_{F}\equiv {\frac {N_{c}^{2}-1}{2N_{c}}}={\frac {4}{3}}}$ , where ${\displaystyle N_{c}=3}$  is the number of color charges.

The value for ${\displaystyle \alpha _{s}}$  depends on the radius of the studied hadron. Its value ranges from 0.19 to 0.4.^[4] For precise determination of the short distance potential, the running of ${\displaystyle \alpha _{s}}$  must be accounted for, resulting in a distant-dependent ${\displaystyle \alpha _{s}(r)}$ . Specifically, ${\displaystyle \alpha _{s}}$  must be calculated in the so-called potential renormalization scheme (also denoted V-scheme) and, since quantum field theory calculations are usually done in momentum space, Fourier transformed to position space.^[4]

The second term of the potential, ${\displaystyle \sigma \,r}$ , is the linear confinement term and fold-in the non-perturbative QCD effects that result in color confinement. ${\displaystyle \sigma }$  is interpreted as the tension of the QCD string that forms when the gluonic field lines collapse into a flux tube. Its value is ${\displaystyle \sigma \sim 0.18}$  GeV${\displaystyle ^{2}}$ .^[4] ${\displaystyle \sigma }$  controls the intercepts and slopes of the linear Regge trajectories.

The Cornell potential applies best for the case of static quarks (or very heavy quarks with non-relativistic motion), although relativistic improvements to the potential using speed-dependent terms are available.^[3] Likewise, the potential has been extended to include spin-dependent terms^[3]

A test of validity for approaches that seek to explain 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 is to be able compute from first principles the static quark-antiquark potential, with results confirming the empirical Cornell Potential.^[5]

Other approaches to the confinement problem also results in the Cornell potential, including the dual superconductor model, the Abelian Higgs model, the center vortex models.^[3][6]

More recently, calculations based on the AdS/CFT correspondence have reproduced the Cornell potential using the AdS/QCD correspondence^[7][8] or light front holography.^[9]

• Color confinement
• QCD vacuum