Hadron Loops: General Theorems and Application to Charmonium

In this paper we develop a formalism for incorporating hadron loops in 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 ${L}_i eq L_f$ or $S_i eq S_f$. The charmonium system is used as a numerical case study, with the $^3P_0$ 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$^*$, D$_s$ and D$_s^*$ meson pairs. We find that the mass shifts are quite large, but are 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.