Thom-Sebastiani theorem in the motivic world

Sebastiani and Thom proved, in the complex analytic setting, that the vanishing cycle of f+g is the product of the vanishing cycles of f and g. In the motivic world, Deligne has observed that such a formula cannot hold, unless one considers instead the convolution product with respect to the monodromy. This formula was then obtained for étale sheaves by Illusie, and for mixed Hodge modules by Saito. We will present here a proof working for Morel-Voevodsky motivic stable homotopy theory (SH), using deep results of Ayoub. Passing to the Grothendieck group, one obtains the Thom-Sebastiani theorem of Denef-Loeser-Looijenga.
We will then explain how, using this result, vanishing cycles of quadratic forms give square roots of Thom twists, generalizing the results of Deligne for étale sheaves. We will then sketch how these results could be used to upgrade cohomological DT theory at the level of SH, up to some important orientation issues.