  Around 2008 N. Kawazumi and S. Zhang introduced a new fundamental numerical
invariant for compact Riemann surfaces. One way of viewing the Kawazumi-Zhang
invariant is as a quotient of two natural hermitian metrics with the same first
Chern form on the line bundle of holomorphic differentials. In this paper we
determine precise formulas, up to and including constant terms, for the
asymptotic behavior of the Kawazumi-Zhang invariant for degenerating Riemann
surfaces. As a corollary we state precise asymptotic formulas for the
beta-invariant introduced around 2000 by R. Hain and D. Reed. These formulas
are a refinement of a result Hain and Reed prove in their paper. We illustrate
our results with some explicit calculations on degenerating genus two surfaces.
