Harmonic oscillator force between heavy quarks

Abstract

A renormalization-group procedure for effective particles is applied to quantum chromodynamics of one flavor of quarks with a large mass m in order to calculate light-front Hamiltonians for heavy quarkonia ${H}_{\ensuremath{\lambda}}$ using perturbative expansion in the coupling constant ${\ensuremath{\alpha}}_{\ensuremath{\lambda}}.$ $\ensuremath{\lambda}$ is the renormalization-group parameter with the interpretation of an inverse of the spatial size of the color charge distribution in the effective quarks and gluons. The eigenvalue equation for ${H}_{\ensuremath{\lambda}}$ couples quark-antiquark states with sectors of a larger number of constituents. The coupling to states with more than one effective gluon, and interactions in the quark-antiquark-gluon sector, are removed at the price of introducing an ansatz for the gluon mass ${\ensuremath{\mu}}^{2}.$ The simplified equation is used to evaluate a new Hamiltonian of order ${\ensuremath{\alpha}}_{\ensuremath{\lambda}}$ that acts only in the effective quark-antiquark sector and in the nonrelativistic limit turns out to contain the Coulomb term with Breit-Fermi corrections and a spin-independent harmonic oscillator term with a frequency $\ensuremath{\omega}=[(4/3)({\ensuremath{\alpha}}_{\ensuremath{\lambda}}/\ensuremath{\pi}){]}^{1/2}\ensuremath{\lambda}(\ensuremath{\lambda}{/m)}^{2}(\ensuremath{\pi}{/1152)}^{1/4}.$ The latter originates from the hole excavated in the overlapping quark self-interaction gluon clouds by the exchange of effective gluons between the quarks. The new term is largely independent of the details of ${\ensuremath{\mu}}^{2}$ and in principle can fit into the ballpark of phenomenology. The first approximation can be improved by including more terms in ${H}_{\ensuremath{\lambda}}$ and solving the eigenvalue equations numerically.