Hadron loops: General theorems and application to charmonium

In this paper, we develop a formalism for incorporating hadron loops into the quark model. We derive expressions for mass shifts, continuum components, and mixing amplitudes of ``quenched'' quark model states due to hadron loops, as perturbation series in the valence-continuum coupling Hamiltonian. We prove three general theorems regarding the effects of hadron loops, which show that given certain constraints on the external ``bare'' quark model states, the valence-continuum coupling, and the hadrons summed in the loops, the following results hold: (1) The loop mass shifts are identical for all states within a given $N,L$ multiplet. (2) These states have the same total open-flavor decay widths. (3) Loop-induced valence configuration mixing vanishes provided that Lᵢ≠Lf or Sᵢ≠Sf. The charmonium system is used as a numerical case study, with the ³P₀ decay model providing the valence-continuum coupling. We evaluate the mass shifts and continuum mixing numerically for all $1S,1P$, and $2S$ charmonium valence states due to loops of D,D*,Dₛ, and Dₛ* meson pairs. We find that the mass shifts are quite large but numerically similar for all the low-lying charmonium states, as suggested by the first theorem. Thus, loop mass shifts may have been ``hidden'' in the valence quark model by a change of parameters. The two-meson continuum components of the physical charmonium states are also found to be large, creating challenges for the interpretation of the constituent quark model.