Complex chromoelectric polarizability of a heavy-quarkonium resonance: pole response and channel-resolved effective-theory representation

For a stable compact quarkonium state in the pNRQCD multipole regime, chromoelectric polarizability arises at leading order from two E1 transitions through virtual color-octet states. For an unstable state, we define the corresponding response through the quadratic displacement of its complex resonance pole in a weak chromoelectric background, so that its real and imaginary parts describe field-induced shifts of the pole mass and width. Starting from a renormalized background-field correlator, we show that the resonance polarizability is given by the complete two-field kernel normalized by the energy derivative of the full inverse propagator at the pole. In an effective theory with explicitly separated intermediate sectors, the kernel contains the standard E1-octet-E1 contribution, the complete source response of decay channels already coupled at zero field, and local contributions. This distinction becomes essential above open-flavor threshold and reduces to the standard pNRQCD polarizability in the stable, energy-independent limit. A simple $D\bar{D}$ realization for the $\psi (3770)$ gives the complex pole-residue factor $1\mathrm{/}[1-{\sigma }^{\mathrm{\prime }}(z_{\psi })]=0.800-0.336$, illustrating that threshold dressing can substantially modify both the magnitude and phase of the response. The resulting pole-based relation identifies the quantity required for a source-dependent coupled-channel determination of the chromoelectric polarizability of a heavy-quarkonium resonance.