The chromoelectric polarizability of an unstable quarkonium is the complex-pole curvature equal to the complete two-field vertex divided by the energy derivative of the inverse propagator, recovering the stable-state result in the narrow limit.
Long-range properties of $1S$ bottomonium states
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abstract
In the framework of weakly-coupled pNRQCD, we derive, first, an analytical expression for the chromo-polarizability of $1S$-bottomonium states in agreement with previous determinations. Then we use the QCD trace anomaly to obtain the two-pion production amplitude for the chromo-polarizability operator and match the result to a chiral effective field theory with $1S$-bottomonium states and pions as degrees-of-freedom. In this chiral effective field theory we compute some long-range properties of the $1S$ bottomonium generated by the pion coupling such as the leading chiral logarithm to the $1S$-bottomonium mass and the van der Waals potential between two $1S$-bottomonium states. Both results improve on previously known expressions.
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Complex chromoelectric polarizability of a heavy-quarkonium resonance: pole definition and channel-complete pNRQCD matching
The chromoelectric polarizability of an unstable quarkonium is the complex-pole curvature equal to the complete two-field vertex divided by the energy derivative of the inverse propagator, recovering the stable-state result in the narrow limit.