This paper concerns a method of describing hadrons that starts with the canonical front form Hamiltonian of QCD. The method is developed in the relatively simple context of QCD with only heavy quarks. We regulate its canonical Hamiltonian by introducing a vanishingly small gluon mass $m_g$. For positive $m_g$, the small-$x$ gluon divergences become ultraviolet and hence they are renormalized in the same way the ultraviolet transverse divergences are. This is done using the renormalization group procedure for effective particles. Up to the second order of expansion of the renormalized Hamiltonian in powers of the quark-gluon coupling constant $g$, only the quark mass-squared and gluon-exchange divergences require counterterms. In these circumstances, we calculate an effective potential between quarks in heavy quarkonia in an elementary way, replacing all the quarkonium-state components with gluons of mass $m_g$ by only one component with just one gluon that is assigned a mass $m_G$, comparable to or exceeding the scale of typical relative momenta of bound quarks. In the limit of $m_g\to 0$ and large $m_G$ two results are obtained. (1) While the color-singlet quarkonium mass eigenvalue stays finite and physically reasonable in that limit, the eigenvalues for single quarks and octet quarkonia are infinite. (2) Besides the coulomb terms, the effective quark-antiquark potential is quadratic as a function of the distance and spherically symmetric for typical separations between quarks but becomes logarithmic and no longer spherically symmetric for large separations. Our conclusion indicates how to systematically improve upon the approximations made in this paper.

• Received 7 October 2023
• Accepted 29 November 2023

DOI:https://doi.org/10.1103/PhysRevD.109.016017

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Published by the American Physical Society