Renormalon Saddles in the OPE

Abstract: The cancellation of renormalon ambiguities in the operator product expansion is usually formulated as a matching between perturbative Borel ambiguities and prescription dependence in non-perturbative condensates. We give this cancellation a contour realization in the two-dimensional Gross--Neveu model at large $N$. The exact order-$1/N$ fermion self-energy in the massive vacuum is a finite real integral; ambiguities arise only after its large-momentum behavior is decomposed into products of perturbative Wilson coefficients and the matrix elements of the associated operators. A reduced integral isolates the leading renormalon and reveals a two-dimensional contour geometry in which the perturbative series is generated at the boundary corner where the exchanged momentum reaches the external hard scale, while its Stokes discontinuity is controlled by a distinct boundary critical point at zero exchanged momentum. After renormalizing the ultraviolet divergences of the condensate scale integral, we derive its compensating lateral ambiguity without fixing it by matching to the perturbative sector, distinguish the renormalon saddle from the apparent IR Landau pole, and obtain the reduced contour variables from the large-$N$ auxiliary-field path integral. The renormalon saddle is a boundary critical point of this reduced mode-space integral. The perturbative Wilson coefficient and the renormalized condensate thus define complementary relative cycles of the same reduced integrand whose complex tails cancel, reconstructing the original real integration cycle.