Structural aspects of FRG in quantum tunnelling computations

TL;DR

This paper uses the Functional Renormalization Group to analyze quantum tunneling in the quartic oscillator and double well potential, assessing the method's accuracy across different coupling regimes.

Contribution

It provides a numerical study of FRG flow equations truncated at first order, evaluating their effectiveness in non-perturbative quantum tunneling problems.

Findings

Accurate results in high lambda regime with first-order approximation.

Good agreement with exact results in intermediate lambda range.

Higher order corrections needed for small lambda regime.

Abstract

We probe both the unidimensional quartic harmonic oscillator and the double well potential through a numerical analysis of the Functional Renormalization Group flow equations truncated at first order in the derivative expansion. The two partial differential equations for the potential V_k(varphi) and the wave function renormalization Z_k(varphi), as obtained in different schemes and with distinct regulators, are studied down to k=0, and the energy gap between lowest and first excited state is computed, in order to test the reliability of the approach in a strongly non-perturbative regime. Our findings point out at least three ranges of the quartic coupling lambda, one with higher lambda where the lowest order approximation is already accurate, the intermediate one where the inclusion of the first correction produces a good agreement with the exact results and, finally, the one with smallest lambda where presumably the higher order correction of the flow is needed. Some details of the specifics of the infrared regulator are also discussed.