Report
From the very beginning, in the introduction, the authors emphasize the so-called technical difficulties in the study of the conspiracy between renormalons and OPE, the absence of the manifest O(N) invariance in their calculations, and so on. As a result, they come to a paradoxical conclusion that the ambiguity in a vacuum expectation value conspires with the *ultraviolet* renormalon rather than the IR one. Why it is paradoxical? UV renormalons can be uniquely summed up through the Borel procedure. I believe that their difficulties are mostly self-made, because the calculation method they chose is awkward (and was known to be awkward from the 1980s). This is seen, for instance, from their Eqs. (2.10) and (2.11) demonstrating the need to calculate separately the beta function and the anomalous dimension of the sigma filed. Also, the loss of the explicit O(N) invariance points in the same direction. The controversial part is formulated after the Fig. 4. To my mind, the problem is that the authors calculate the self-energy of the sigma field, which is unobservable. In this model only correlates of O(N) symmetric operators are to be considered, much in the same way as in Yang-Mills theory, if one considers gauge non-invariant operators en route, it is indeed difficult to avoid misinterpretations. If they used the background field formalism, the O(N) symmetry of the model would be maintained explicitly, and therefore it would be obvious that there is only one constant to be calculated, the overall coupling constant, which would lead to the beta functions. They refer to ref. [30] at this point which did not address the issue of conspiracy. In view of the conceptual importance of the statement of UV renormalon/condesate conspiracy advocated by the authors, I think this aspect cannot be dealt with just in passing. It goes against all foundations of OPE and the conspiracy phenomenon. If the authors insist on this interpretation, they should at least add a very detailed general analysis how it can be made compatible with OPE or, better still, split the paper in twi parts – general foundation and concrete calculations. A marginal reference [30] is by far insufficient. Summarizing, I cannot recommend this paper for publication.
