A Simple Approach to the Charmonium Spectrum
August 01, 2022
In this work we derive an analytical mass formula for the charmonium states. Starting from the two-body Hamiltonian with the Cornell potential, we use a version of the uncertainty relation (which depends on the $n$ and $l$ quantum numbers) and write the momentum in terms of the relative coordinate. Minimizing the obtained Hamiltonian with respect to this coordinate we determine the equilibrium separation and calculate the energy of the state, which depends on the charm mass, on the coupling constant and on the string tension. We include the spin interactions. These parameters are fixed through a fit of the existing data. We obtain a good fit of the data. This mass formula can be useful to make a first estimate of new states and can also be easily adapted to multiquark systems such as the X,Y and Z states.
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