Hadron Loops: General Theorems and Application to Charmonium

TL;DR

This paper develops a formalism to incorporate hadron loops into the quark model, deriving theorems about their effects on mass shifts, decay widths, and mixing, with applications to charmonium states.

Contribution

It introduces a general formalism and proves theorems on hadron loop effects, applying them to charmonium to analyze mass shifts and continuum components.

Findings

Mass shifts are similar across low-lying charmonium states.

Large continuum components are present in physical states.

Loop effects can be hidden in quark model parameters.

Abstract

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 ${\L}_i \neq \L_f$ or $\S_i \neq \S_f$. The charmonium system is used as a numerical case study, with the $^3\P_0$ 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.