The regulator dependence in the functional renormalization group: a quantitative explanation

Abstract

The search of controlled approximations to study strongly coupled systems remains a very general open problem. Wilson's renormalization group has shown to be an ideal framework to implement approximations going beyond perturbation theory. In particular, the most employed approximation scheme in this context, the derivative expansion, was recently shown to converge and yield accurate and very precise results. However, this convergence strongly depends on the shape of the employed regulator. In this letter we clarify the reason for this dependence and justify, simultaneously, the most largely employed procedure to fix this dependence, the principle of minimal sensitivity.

Local Disontinuous Galerkin for the Functional Renormalisation Group

“…Within the fRG, it has been studied extensively [5,[8][9][10][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41], ranging from fixed points, to its RG-time evolution in physical systems. Therefore, it constitutes an excellent toy-model for our purpose.…”

“…There is no spontaneous symmetry breaking process and the result for a simple φ 4 theory can also be obtained from a numerical evaluation of the path integral (1). Thus, the full effective potential V k=0 (ρ) is computed in d = 0 by a direct numerical evaluation of Z[J] with the classical action specified by the initial conditions in (41), following the steps of the derivation in Section II. In order to capture the full result with the fRG, the computation is started at an initial cutoff scale of Λ = 20, where the numerical result of the modified path integral Z k [J] (3) corresponds to the initial conditions V Λ (ρ) (41).…”

“…5: Graphical comparison of the purely convective large-N limit and the convection-diffusion system (36) at N = 100. We consider the broken symmetry phase in d = 3 using the initial conditions from (41). In the large-N limit, the derivative is obtained by interpolation of the numerical data.…”