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

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.