Feasibility of Diagrammatic Monte-Carlo based on weak-coupling expansion in asymptotically free theories: case study of $O(N)$ sigma-model in the large-$N$ limit

Abstract

We discuss the feasibility of applying Diagrammatic Monte-Carlo algorithms to the weak-coupling expansions of asymptotically free quantum field theories, taking the large-$N$ limit of the $O(N)$ sigma-model as the simplest example where exact results are available. We use stereographic mapping from the sphere to the real plane to set up the perturbation theory, which results in a small bare mass term proportional to the coupling $\lambda$. Counting the powers of coupling associated with higher-order interaction vertices, we arrive at the double-series representation for the dynamically generated mass gap in powers of both $\lambda$ and $\log (\lambda )$, which converges quite quickly to the exact non-perturbative answer. We also demonstrate that it is feasible to obtain the coefficients of these double series by a Monte-Carlo sampling in the space of Feynman diagrams. In particular, the sign problem of such sampling becomes milder at small $\lambda$, that is, close to the continuum limit.

Comments: 15 pages PoS style, 2 figures. Extended version of the contribution to the Proceedings of the Lattice 2015 Conference (14-18 July 2015, Kobe, Japan). Also partially includes the material of the talks given at the Sign 2015 conference (28 September - 2 October 2015, Debrecen, Hungary) and the ECT* workshop on Diagrammatic Monte-Carlo (5-9 October 2015, ECT*, Trento, Italy)