Abstract
We study the $SU(2)$ Principal Chiral Model (PCM) in the presence of an integrable $\eta$-deformation. We put the theory on $\mathbb{R}\times S^1$ with twisted boundary conditions and then reduce the circle to obtain an effective quantum mechanics associated with the Whittaker-Hill equation. Using resurgent analysis we study the large order behaviour of perturbation theory and recover the fracton events responsible for IR renormalons. The fractons are modified from the standard PCM due to the presence of this $\eta$-deformation but they are still the constituents of uniton-like solutions in the deformed quantum field theory. We also find novel $SL(2,\mathbb{C})$ saddles, thus strengthening the conjecture that the semi-classical expansion of the path integral gives rise to a resurgent transseries once written as a sum over Lefschetz thimbles living in a complexification of the field space. We conclude by connecting our quantum mechanics to a massive deformation of the $\mathcal{N}=2~$ $4$-d gauge theory with gauge group $SU(2)$ and $N_f=2$.
Highlights
1.1 The resurgence paradigmThe calculation of the anomalous magnetic moment of the electron to ten significant figures stands as testament to efficacy of perturbation theory in quantum field theory (QFT) [1]
We study the SU(2) Principal Chiral Model (PCM) in the presence of an integrable η-deformation
We find novel SL(2, C) saddles, strengthening the conjecture that the semi-classical expansion of the path integral gives rise to a resurgent transseries once written as a sum over Lefschetz thimbles living in a complexification of the field space
Summary
The calculation of the anomalous magnetic moment of the electron to ten significant figures stands as testament to efficacy of perturbation theory in quantum field theory (QFT) [1]. If B[Epert](t) has singularities in the complex wedge − ≤ arg( t) ≤ , these two lateral summations, S± [Epert](g), are generically different but still yield the same asymptotic perturbative expansion (1.1) once expanded at weak coupling This jumping as one deforms an integration cycle is known as the Stokes phenomenon. As we vary the argument of the coupling constant two things will happen: first the semi-classical expansion will generically receive contributions from all finite action complex saddles living in a complexification of the space of fields and secondly precisely at Stokes line this thimble decomposition will jump discontinuously, that is, in addition to the jump in the resummation of the perturbative series, the sum over non-perturbative saddle points will jump and the two “ambiguities” will precisely cancel. After we sum over all the finite action complexified saddles, the semiclassical expansion (1.5) should not be thought of as an approximation, a resurgent trans-series is an exact coded representation of the physical observable, i.e. it really is an analytic function in a certain wedge of the complex coupling plane
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