The $\imath ε$ prescription in the SYK model

TL;DR

This work introduces an $i\epsilon$ regularization for the SYK model to regulate UV divergences that arise in the conformal IR, providing a universal framework for studying departures from conformality. It defines a regulated two-point function $G^{\epsilon}_{\beta}$ that acts as a frequency cutoff and proves that the induced kernel $A^{\epsilon}_{\beta}$ converges to the delta distribution as $\epsilon\to 0$, enabling a rigorous Schwinger-Dyson formulation at large-$N$. Building on this, the authors construct an infrared-effective field theory with a bi-local counterterm $A^{\epsilon}_{\beta}$ so that the melonic resummation reproduces $G^{\epsilon}_{\beta}$, with a well-defined bare covariance $G^{\epsilon}_{\beta}/(1 - A^{\epsilon}_{\beta})$ for sufficiently large $\epsilon$. The framework yields a controlled, UV-regularized description of SYK IR physics and clarifies how conformal invariance is broken by the regulator while guaranteeing a smooth restoration in the $\epsilon\to 0$ limit.

Abstract

We introduce an $\imath ε$ 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 $ε\to 0$ 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 $\imath ε$ prescription we introduce an effective field theory Lagrangian for the infrared SYK model.

Theorems & Definitions (10)

• Theorem 1
• proof
• Theorem 2
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof