The two - phase issue in the O ( n ) non - linear σ - model : a Monte - Carlo study

The two-phase issue in the O(n) non-linear σ-model: a Monte-Carlo study Our paper [1] has motivated a comment [2] by A. Patrascioiu and E. Seiler which we reply in this note. The remarks in [2] concern three statements that the authors select from our paper: i) " The results for O(8) support the asymptotic freedom scenario ". The authors in [2] recall that at small˜β ≡ β/N the 1/N expansion is Borel summable [3]. No rigorous proof has been given concerning the behaviour of the series at large˜β. In [2] it is not shown if our working β's (β ∼ 4.6 − 6.5) lie in the " small " or " large " ˜ β region when N = 8 but at least we can say that if the results from our Monte Carlo data fit so nicely the P T predictions for the O(8) model (within few per mille for the mass gap and susceptibility predictions, see [4]) there is reasonable room to think that the exact O(8) model (what we simulate) has a critical point at g = 0 and the set of P T predictions are correct. If our data had to agree with some prediction other than the P T set of predictions for the O(8) model then the small difference between our result for the mass gap and the P. Hasenfratz et al. [5] prediction, 0.5% (compared for instance to the difference of almost 30% between the n = 8 and the n = ∞ predictions) would become an intriguing challenge. Considering it as an accident is a matter of feelings.