Bion non-perturbative contributions versus infrared renormalons in two-dimensional $\mathbb C P^{N-1}$ models

TL;DR

This paper calculates non-perturbative bion contributions in the two-dimensional P^{N-1} model, revealing their relation to infrared renormalons and showing how these effects depend on supersymmetry and quantum fluctuations.

Contribution

It provides the first explicit derivation of complex bion contributions and their connection to infrared renormalons in the P^{N-1} model using semiclassical and Lefschetz thimble methods.

Findings

Non-perturbative bion contributions are derived explicitly.

In supersymmetric cases, the contributions vanish.

In non-supersymmetric cases, contributions have an imaginary ambiguity.

Abstract

We derive the semiclassical contributions from the real and complex bions in the two-dimensional $\mathbb C P^{N-1}$ sigma model on ${\mathbb R} \times S^{1}$ with a twisted boundary condition. The bion configurations are saddle points of the complexified Euclidean action, which can be viewed as bound states of a pair of fractional instantons with opposite topological charges. We first derive the bion solutions by solving the equation of motion in the model with a potential which simulates an interaction induced by fermions in the $\mathbb C P^{N-1}$ quantum mechanics. The bion solutions have quasi-moduli parameters corresponding to the relative distance and phase between the constituent fractional instantons. By summing over the Kaluza-Klein modes of the quantum fluctuations around the bion backgrounds, we find that the effective action for the quasi-moduli parameters is renormalized and becomes a function of the dynamical scale (or the renormalized coupling constant). Based on the renormalized effective action, we obtain the semiclassical bion contribution in a weak coupling limit by making use of the Lefschetz thimble method. We find that the non-perturbative contribution vanishes in the supersymmetric case and it has an imaginary ambiguity which is consistent with the expected infrared renormalon ambiguity in non-supersymmetric cases. This is the first explicit result indicating the relation between the complex bion and the infrared renormalon.