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Renormalization group for effective field theories: cutoff schemes and universality

Jose Gaite

TL;DR

This work analyzes universality and regulator-dependence in a three-dimensional scalar EFT with $(\lambda\phi^4+g\phi^6)_3$. It contrasts Wilsonian exact RG calculations with two-loop perturbation theory under general cutoff functions, tracking the flows of $m$, $\lambda$, and $g$ with a finite cutoff $\Lambda_0$ and defining the dimensionless ratio $u=\lambda/m$. The main finding is that the pure quartic theory is universal, while a nonzero sextic coupling introduces non-universal corrections that depend on the regulator through constants $F_1$, $F_2$, $A$, $B$; these corrections are smallest for the sharp cutoff. The authors develop an improved perturbation framework via two-loop beta functions $\beta_1,\beta_2$ and RG-improved trajectories, which agree with exact RG results for small $u$ and illuminate the crossover from tricritical to critical behavior near the Wilson-Fisher point. The results provide practical guidance on regulator choices and quantify regulator-induced effects in EFTs relevant to statistical physics and beyond-Standard-Model applications.

Abstract

In effective field theories, the concept of renormalization of perturbative divergences is replaced by renormalization group concepts such as relevance and universality. Universality is related to cutoff scheme independence in renormalization. Three-dimensional scalar field theory with just the quartic coupling is universal but the less relevant sextic coupling introduces a cutoff scheme dependence, which we quantify by three independent parameters, in the two-loop order of perturbation theory. However, reasonable schemes only allow reduced ranges of those parameters, even contrasting the sharp cutoff with very smooth cutoffs. The sharp cutoff performs better. In any case, the effective field theory possesses some degree of universality even in the massive case (off criticality).

Renormalization group for effective field theories: cutoff schemes and universality

TL;DR

This work analyzes universality and regulator-dependence in a three-dimensional scalar EFT with . It contrasts Wilsonian exact RG calculations with two-loop perturbation theory under general cutoff functions, tracking the flows of , , and with a finite cutoff and defining the dimensionless ratio . The main finding is that the pure quartic theory is universal, while a nonzero sextic coupling introduces non-universal corrections that depend on the regulator through constants , , , ; these corrections are smallest for the sharp cutoff. The authors develop an improved perturbation framework via two-loop beta functions and RG-improved trajectories, which agree with exact RG results for small and illuminate the crossover from tricritical to critical behavior near the Wilson-Fisher point. The results provide practical guidance on regulator choices and quantify regulator-induced effects in EFTs relevant to statistical physics and beyond-Standard-Model applications.

Abstract

In effective field theories, the concept of renormalization of perturbative divergences is replaced by renormalization group concepts such as relevance and universality. Universality is related to cutoff scheme independence in renormalization. Three-dimensional scalar field theory with just the quartic coupling is universal but the less relevant sextic coupling introduces a cutoff scheme dependence, which we quantify by three independent parameters, in the two-loop order of perturbation theory. However, reasonable schemes only allow reduced ranges of those parameters, even contrasting the sharp cutoff with very smooth cutoffs. The sharp cutoff performs better. In any case, the effective field theory possesses some degree of universality even in the massive case (off criticality).

Paper Structure

This paper contains 20 sections, 81 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Renormalization of scalar field theory in three-dimensional space
  3. Results of the $(\lambda\phi^4)_3$ theory
  4. Exact renormalization group calculations for fixed bare coupling
  5. Two-loop perturbative $(\lambda\phi^4+g\phi^6)_3$ theory
  6. Calculation of non-universal corrections
  7. Renormalization with $g_0 \neq 0$ and sharp cutoff
  8. Renormalization with $g_0 \neq 0$ in various schemes
  9. Hyperbolic tangent
  10. Exponential function
  11. Power-law function
  12. Conclusion of the examination
  13. Improved perturbation theory
  14. Improving $(\lambda\phi^4+g\phi^6)_3$ perturbation theory
  15. Conclusions
  16. ...and 5 more sections

Figures (3)

  • Figure 1: The three types of cutoff functions and the corresponding values of $F_1$ plotted against cutoff function widths. First row: hyperbolic tangent (plotted for $\varepsilon=1,1/5$); second row: exponential (plotted for $n=2,7$); third row: power law (plotted for $n=2,10$).
  • Figure 2: RG flow for decreasing mass (in sharp-cutoff renormalization). Several features described in the text are displayed. Light red and black points correspond to Wegner-Houghton RG and two-loop calculations, respectively, nearly coinciding within some range (red points on top). The furthest couple of points has $m/\Lambda_0=0.01$.
  • Figure 3: Scheme dependence of perturbative results from Eqs. (\ref{['l0lr2loop']}) and (\ref{['gg0l2loop']}) (interpolated). Sharp cutoff: lower (blue) line. Smooth $n=2$ power-law cutoff: upper (magenta) line. The two couples of points correspond to $m/\Lambda_0=0.71$ and $m/\Lambda_0=0.12$, respectively.