Low energy limit from high energy expansion in mass gapped theory

We present a method to extract the low energy behavior of physical observables from their high energy expansions, systematically calculable via the operator product expansion (OPE), in asymptotically free and mass-gapped theories. By applying the inverse Laplace transform to correlation functions, their analytic structure is modified such that low-energy information connects with high energy expansions. Furthermore, this transformation alleviates the renormalon problem, enabling a more straightforward application of the OPE compared to the OPE before the transformation. We demonstrate that the low energy limit of correlation functions can be accurately extracted using the OPE in the two dimensional $O(N)$ nonlinear $\sigma$ model, serving as a first testing ground.