Integration of Identity with Divergent Operator

Question

I would like to know what results in the integration of the identity below

I think the left side is what gives $\varphi(\theta)$, using the Divergence Theorem, but I'm in doubt because there's an $\xi$ term in $\nabla\cdot (\exp(-\pi e^{i \theta}|\xi|^2)\xi)$, then I don't know if there's anything else to cancel with a term on the right side.