Complex chromoelectric polarizability of a heavy-quarkonium resonance: pole definition and channel-complete pNRQCD matching

Abstract

For a stable compact quarkonium, chromoelectric polarizability is generated by two E1 transitions through virtual color-octet states. An unstable quarkonium is instead defined by an isolated complex pole. We define its polarizability by the quadratic displacement of that pole. Combining the standard pole-residue definition of resonance properties with pNRQCD matching shows that the complete two-field vertex is normalized by the energy derivative of the full inverse propagator. Its numerator contains octet E1-E1 propagation, the field dependence of decay channels, and local hard matching terms, without double counting. The stable-state result is recovered with its standard sign and color factor. A minimal single-threshold model calibrated to the Psi(3770) pole gives, within its specified field convention, the residue factor 0.800-0.336i. This illustrates a potentially sizable normalization effect but is not a model-independent pole observable. The absolute polarizability still requires quarkonium and channel response inputs.