Abstract
We consider the operator product expansion (OPE) of correlation functions in the supersymmetric $O(N)$ non-linear sigma model at sub-leading order in the large $N$ limit in order to study the cancellation between ambiguities coming from infrared renormalons and those coming from various operators in the OPE. As has been discussed in the context of supersymmetric Yang-Mills theory in four dimensions, supersymmetry presents a challenge to this cancellation. In a bid to solve this problem we consider $O(N)$ as a toy model. A background field method inspired by Polyakov's treatment of the renormalization of the bosonic $O(N)$ model is used to identify explicit operators in the OPE of the two-point functions of bosonic and fermionic fields in the model. In order to identify the coefficient functions in the OPE, the exact two-point functions at sub-leading order in large $N$ are expanded in powers of the natural infrared length scale. The ambiguities arising from renormalons in the coefficient functions and vacuum expectation values of operators in the OPE are shown to cancel to all orders. The question of supersymmetric Yang-Mills theory without matter remains open.
Highlights
In this paper we study the supersymmetric (SUSY) twodimensional OðNÞ model in the ’t Hooft limit [1,2,3,4]
The exact solution teaches us what happens to the divergent perturbation theory and how it conspires with nonperturbative terms in the operator product expansion (OPE)
There will be a host of other dimension-two operators which may pick up the slack, so lower dimension renormalons cannot be ruled out. The presence of these operators to some extent already resolves the difficulty stated above, and the bulk of this paper is about seeing in detail how the cancellation of ambiguities between operators and coefficient functions occurs in the operator product expansion
Summary
In this paper we study the supersymmetric (SUSY) twodimensional OðNÞ model in the ’t Hooft limit [1,2,3,4]. In the nonsupersymmetric OðNÞ model the dimension-two trace of the energy momentum tensor cancels with the lowest renormalon in the identity coefficient function. It might seem that there would be no lower-dimension operators to cancel potential renormalon poles At this point 4D SYM and 2D OðNÞ diverge. The question of the existence of lower-dimension renormalon poles has been investigated in recent works considering the SUSY OðNÞ model with a chemical potential [11,12]. The bona fide verification of conspiracy between renormalons and VEVs of relevant operators in OPE can only be carried out in the abovementioned cases This is our main goal in the present paper. The best one can do in the case μ ≫ Λ is to check that the artificial parameter μ drops out of OPE [6]
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