Abstract
We derive the semiclassical contributions from the real and complex bions in the two-dimensional ℂPN − 1 sigma model on ℝ×S1 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 ℂPN − 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 in the supersymmetric case that the bion contribution vanishes as expected from supersymmetry. In non-supersymmetric cases, the non-perturbative contribution has an imaginary ambiguity which is consistent with the expected infrared renormalon ambiguity. Our results explicitly demonstrate that the complex bion can explain the infrared renormalon.
Highlights
The “infrared renormalon” observed in the perturbative expansion in QFT [1, 2] is believed to be related to non-perturbative phenomena
In the CP N−1 quantum mechanics corresponding to the small compactification radius limit of the CP N−1 model on R×S1 with the twisted boundary condition, it was shown that the semiclassical contributions from the bion saddle points of the complexified action cancel the imaginary ambiguity in the Borel resummation of the perturbation series
We have calculated the semiclassical contributions from the bion saddle points in the CP N−1 models on R × S1 with twisted boundary conditions, with emphasis on its consistency with the infrared renormalon
Summary
In the present and sections, we investigate bions in the 2D CP 1 sigma model on R×S1 with emphasis on their relevance to the renormalon problem. This model admits 1/2 Bogomol’nyi-Prasad-Sommerfield (BPS) instanton solutions satisfying the BPS equation ∂ ̄φ = 0 [123]. Of the 2D CP 1 model with the twisted boundary condition by evaluating the action for the lightest mode in the KK expansion φ(x, y) = φ0 eimy with constant φ0 This potential exhibits two degenerate discrete minima at φ0 = 0 (north pole) and φ0 = ∞ (south pole), in contrast to the vacuum manifold CP 1 of the untwisted model. Instead of dealing with such an approximate solution of bion, we introduce a deformation parameter so that the equation of motion admits an exact bion solution
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