Tunneling with physics-informed RG flows in the anharmonic oscillator

We solve the anharmonic oscillator with physics-informed renormalisation group (PIRG) flows, with an emphasis on the weak coupling regime with its instanton-dominated tunnelling processes. We show that the instanton physics behind the exponential decay of the energy gap is already covered in the first order of the derivative expansion of the PIRG. The crucial new ingredients in the present analysis are the use of the ground state expansion within PIRG flows, as well as precision numerics based on Galerkin methods. Our result $a_{\mathrm{inst}} = 1.910(2)$ for the decay constant is in quantitative agreement with the analytic one, $a_{\mathrm{inst}} = 1.886$ with a deviation of $1\%$. This illustrates very impressively the capacity of the PIRG for fully capturing non-perturbative physics already in relatively simple approximations.