Long-range properties of $1S$ bottomonium states

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.