A Wick rotation for EPRL spin foam models

We show that the Euclidean and Lorentzian EPRL vertex amplitudes of covariant Loop Quantum Gravity are related through a ``Wick rotation'' of the real Immirzi parameter to purely imaginary values. Our result follows from the simultaneous analytic continuation of the algebras, group elements and unitary irreducible representations of the gauge groups $Spin(4)$ and $SL(2,\mathbb{C})$, applied to the decomposition of the two models in terms of $SU(2)$ invariants and booster functions.