Long-range properties of $1S$ bottomonium states

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.