# Renormalons in quantum mechanics

## TL;DR

This paper demonstrates the existence of renormalon divergences in a specific nonrelativistic quantum mechanics model with delta-function potentials, showing how they affect perturbative series and are resolved by causality conditions.

## Contribution

It explicitly computes renormalon divergences in a quantum mechanics model and shows their general presence in similar potentials, connecting perturbative ambiguities with physical causality.

## Key findings

- Renormalons appear in the perturbative S-matrix of the model.
- Borel summation ambiguity is resolved by causality and the $i\,\epsilon$ prescription.
- Perturbative results match the exact operator formalism solutions.

## Abstract

We present a nonrelativistic one-particle quantum mechanics whose perturbative S-matrix exhibits a renormalon divergence that we explicitely compute. The potential of our model is the sum of the 2d Dirac $\delta$-potential -- known to require renormalization -- and a 1d Dirac $\delta$-potential tilted at an angle. We argue that renormalons are not specific to this example and exist for a much wider class of potentials. The ambiguity in the Borel summation of the perturbative series due to the renormalon pole is resolved by the physical condition of causality through careful consideration of the $i\epsilon$ prescription. The suitably summed perturbative result coincides with the exact answer obtained through the operator formalism for scattering.