The $\imath \epsilon$ prescription in the SYK model

We introduce an $\imath \epsilon$ prescription for the SYK model both at finite and at zero temperature. This prescription regularizes all the naive ultraviolet divergences of the model. As expected the prescription breaks the conformal invariance, but the latter is restored in the $\epsilon \to 0$ limit. We prove rigorously that the Schwinger Dyson equation of the resummed two point function at large $N$ and low momentum is recovered in this limit. Based on this $\imath \epsilon$ prescription we introduce an effective field theory Lagrangian for the infrared SYK model.