The Cornell Potential as theTangent of a Stereographic Latitude

AbstractThe Coulomb-plus-linear (Cornell) potential V(r) = −κ/r + σr is, exactly and for every r, anenergy times the tangent of the latitude of a point on a sphere under stereographicprojection. The Coulomb term and the confining term are the two asymptotes of onebranch of the tangent: the south pole (r = 0) and the north pole (r → ∞). The equator sits atℓ = √(κ/σ), where the two forces are equal.Two consequences follow. The potential is odd under the inversion r → ℓ²/r, and in thelogarithmic (Mercator) coordinate it is a hyperbolic sine. The same cotangent of a polarangle read through an equidistant map gives the known finite-range trigonometricextension of the Cornell potential; read through the stereographic map it gives the flatCornell potential with no higher terms.The inversion property was tested on the published continuum static force and potentialof quenched lattice QCD (Necco and Sommer, 2002), over 0.08 to 0.55 fm. With a singleinversion radius ℓ = 0.4255 r₀, the quantity rF(r) is symmetric under r → ℓ²/r to within 1%(rms), and V(r) + V(ℓ²/r) is constant to within 0.6% of the range of V. The only visibledeparture, about 2% at the shortest distance, is in the direction expected from therunning coupling.Results are labelled [A] (derived or checked against data), [B] (hypothesis withmechanism, prediction and a way to fail) and [C] (open conjecture). Keywords: Cornell potential, quark confinement, static quark potential, stereographic projection, Riemann sphere, inversion symmetry, lattice QCD, Coulomb-plus-linear potential, Gudermannian function, string tensionAs cinco primeiras são as que um físico de QCD usaria para buscar o tema, e as cinco últimas carregam o que a sua nota tem de próprio. No campo "Subjects", vale escolher High Energy Physics – Phenomenology, que é onde o potencial de Cornell é discutido.