  Let $L$ be a cyclic $L_\infty$-algebra of dimension $3$ with finite
dimensional cohomology only in dimension one and two. By transfer theorem there
exists a cyclic $L_\infty$-algebra structure on the cohomology $H^*(L)$. The
inner product plus the higher products of the cyclic $L_\infty$-algebra defines
a superpotential function $f$ on $H^1(L)$. We associate with an analytic Milnor
fiber for the formal function $f$ and define the Euler characteristic of $L$ is
to be the Euler characteristic of the \'etale cohomology of the analytic Milnor
fiber.
  In this paper we prove a Thom-Sebastiani type formula for the Euler
characteristic of cyclic $L_\infty$-algebras. As applications we prove the
Joyce-Song formulas about the Behrend function identities for semi-Schur
objects in the derived category of coherent sheaves over Calabi-Yau threefolds.
A motivic Thom-Sebastiani type formula and a conjectural motivic Joyce-Song
formulas for the motivic Milnor fiber of cyclic $L_\infty$-algebras are also
discussed.
