Renormalons in quantum mechanics

Understand

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.