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 δ-potential — known to require renormalization — and a 1d Dirac δ-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 iϵ prescription. The suitably summed perturbative result coincides with the exact answer obtained through the operator formalism for scattering.
Highlights
We present a nonrelativistic one-particle quantum mechanics whose perturbative S-matrix exhibits a renormalon divergence that we explicitely compute
We use the formal tools of quantum mechanical scattering theory to compare the diverging perturbative series to the exact non-perturbative result
We focus on a simple example based on coupling the 2d δ-potential to a 1d δ potential supported along a third direction
Summary
The perturbative series of a physical quantity in a coupling constant λ is often divergent rather than convergent. In this paper we consider the (Lorentzian) S-matrix and the renormalon divergence is of the first type, namely the extra contribution (2.5) from the pole in the Borel plane will not be cancelled by other non-perturbative contributions, but will remain and its sign fixed by the i prescription which is equivalent to the physical condition of causality of scattering. In the case of combinatorially driven growth this is the case [25] and the associated nonperturbative effects are instantons, saddle points in the semi-classical evaluation of the path integral For renormalons such an independent interpretation is not universally established, recently a large effort towards settling this question has been made [26,27,28,29,30,31,32,33,34]. We leave this issue in our model as an interesting open question, noting that the simple and mathematically rigorous setting of quantum mechanics should allow a precise answer to be formulated
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