Second variation of Zhang’s λ-invariant on the moduli space of curves

We compute the second variation of the $\lambda$-invariant, recently introduced by S. Zhang, on the complex moduli space ${\cal M}_g$ of curves of genus $g \geq 2$, using work of N. Kawazumi. As a result we prove that $(8g+4)\lambda$ is equal, up to a constant, to the $\beta$-invariant introduced some time ago by R. Hain and D. Reed. We deduce some consequences; for example we calculate the $\lambda$-invariant for each hyperelliptic curve, expressing it in terms of the Petersson norm of the discriminant modular form.