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| Cornell potential | |
|---|---|
| Name | Cornell potential |
| Type | Interaction potential |
| First proposed | 1970s |
| Common uses | Quarkonium spectroscopy, lattice gauge theory, potential models |
Cornell potential The Cornell potential is a phenomenological two-term potential widely used to model the interaction between heavy quark–antiquark pairs in hadronic physics, combining a short-range Coulomb-like attraction and a long-range linear confinement term. It underpins calculations in heavy quarkonium spectroscopy, informs lattice gauge theory benchmarks, and serves as a starting point in phenomenological treatments within quantum chromodynamics. The potential connects to experimental spectroscopy, numerical simulations, and analytic approximation schemes across particle physics.
The canonical form comprises a Coulombic term plus a linear term: V(r) = -k/r + a r + c, where r denotes the interquark separation and k, a, c are parameters determined from spectroscopy, lattice data, and sum rules. Typical parameter choices reference fits to J/ψ (particle), Υ (particle), ψ(2S), χ_c multiplets and to static potentials extracted in lattice QCD calculations. The Coulombic coefficient k is related to the running coupling in quantum chromodynamics, while the string tension a corresponds to the long-distance flux-tube energy measured in Wilson loop studies and in analyses connected to Regge trajectories.
The short-range -k/r term is motivated by one-gluon exchange computed in perturbative quantum chromodynamics and compared to results from asymptotic freedom and running coupling constant behavior. The linear a r term models confinement consistent with flux-tube pictures, string models, and results from lattice gauge theory simulations of the static potential using Wilson loop and Polyakov loop correlators. Applications include spectroscopy of charmonium, bottomonium, and exotic heavy quark states such as XYZ (particle) resonances, guiding experimental programs at Large Hadron Collider, Belle (particle detector), and BESIII. The potential is also used in studies of quark-gluon plasma probes, nonrelativistic effective field theories like NRQCD, and in comparisons with predictions from AdS/CFT correspondence inspired models.
The Cornell potential is central and central-force problems reduce to a radial Schrödinger equation with angular momentum terms like l(l+1)/r^2; thus solutions involve special functions and semiclassical quantization conditions related to the WKB approximation and to the Bohr–Sommerfeld quantization approach. Spectral properties exhibit discrete bound states for heavy quark masses as in comparisons to the Hydrogen atom spectrum modified by linear confinement; level spacings reflect a competition between Coulombic short-range scaling and linear long-range behavior, connecting to the Virial theorem in bound systems and to scaling relations used in potential scattering theory. The potential leads to nontrivial analytic continuation properties relevant to S-matrix studies and to singularity structure examined in complex scaling methods used in resonance calculations.
Exact analytic solutions do not exist in closed form; instead, one employs numerical integration, variational methods, and approximation schemes such as the WKB approximation, Rayleigh–Ritz method, and perturbation theory around Coulomb or harmonic-oscillator limits. Semirelativistic treatments use the Salpeter equation or the Bethe–Salpeter equation reductions, while effective theories employ matching conditions from NRQCD and pNRQCD. Lattice-extracted potentials constrain parameters used in numerical diagonalization and in basis expansions using Gaussian basis functions, Laguerre polynomials, or harmonic oscillator basis sets. Comparison with experimental decay widths invokes matrix elements computed via potential-model wavefunctions and matched to electromagnetic transition operators measured in collider experiments.
Parameters k, a, and c are obtained by fitting observed masses of heavy quarkonia such as J/ψ (particle), η_c, ψ(3770), Υ(1S), Υ(2S), and fine-structure splittings like those in χ_b and χ_c multiplets. Fits often incorporate inputs from lattice QCD determinations of the static potential and from heavy-quark mass determinations in schemes like MS-bar scheme or pole mass. Systematic uncertainties are assessed by comparing different observables, renormalization prescriptions, and choices of relativistic corrections informed by NRQCD and by radiative corrections computed in perturbative quantum chromodynamics. Experimental programs at CERN, SLAC National Accelerator Laboratory, and KEK provide precision data used in global fits.
Extensions include a screened Cornell form for medium effects relevant to quark–gluon plasma studies at RHIC and LHC Heavy Ion collisions, a relativistic generalization via the Dirac equation with scalar and vector components, and coupled-channel potentials that incorporate open-flavor thresholds such as D meson and B meson channels. Related phenomenological potentials include the Buchmüller–Tye potential, the Logarithmic potential (quarkonium), and potentials motivated by AdS/QCD and by flux-tube models. Comparisons are also made to potentials extracted from lattice calculations of the static interquark force and to effective string models studied by researchers associated with Nambu–Goto action approaches.
Category:Quark models