Stable non-linear evolution in regularised higher derivative effective field theories

TL;DR

This work analyzes a covariant regularisation of higher-derivative EFTs by perturbative field redefinitions that introduce heavy ghost modes while preserving low-energy physics. Focusing on a simple UV Abelian-Higgs-like model, the authors derive EFTs at leading and next-to-leading orders and construct consistently initial data, including for auxiliary variables, to compare with the full UV theory. They show that regularised EFTs (EFT$_2$ and EFT$_4$) admit well-posed evolution and can replicate UV dynamics accurately within the EFT regime, with errors scaling as $M^{-4}$ or $M^{-6}$ and linear-in-time secular growth, in contrast to standard perturbative EFTs which suffer secular blow-up. The results indicate that the regularisation approach provides a robust classical framework for describing UV physics with higher-derivative corrections, and they discuss implications for gravity and global nonlinear stability, as well as practical numerical strategies such as rescaling to improve large-$M$ performance.

Abstract

We study properties of a recently proposed regularisation scheme to formulate the initial value problem for general (relativistic) effective field theories (EFTs) with arbitrary higher order equations of motion. We consider a simple UV theory that describes a massive and a massless scalar degree of freedom. Integrating out the heavy field gives rise to an EFT for the massless scalar. By adding suitable regularising terms to the EFT truncated at the level of dimension-$4$ and dimension-$6$ operators, we show that the resulting regularised theories admit a well-posed initial value problem. The regularised theories are related by a field redefinition to the original truncated EFTs and they propagate massive ghost fields (whose masses can be chosen to be of the order of the UV mass scale), in addition to the light field. We numerically solve the equations of motion of the UV theory and those of the regularised EFTs in $1+1$-dimensional Minkowski space for various choices of initial data and UV mass parameter. When derivatives of the initial data are sufficiently small compared to the UV mass scale, the regularised EFTs exhibit stable evolution in the computational domain and provide very accurate approximations of the UV theory. On the other hand, when the initial gradients of the light field are comparable to the UV mass scale, the effective field theory description breaks down and the corresponding regularised EFTs exhibit ghost-like/tachyonic instabilities. Finally, we also formulate a conjecture on the global nonlinear stability of the vacuum in the regularised scalar EFTs in $3+1$ dimensions. These results suggest that the regularisation approach provides a consistent classical description of the UV theory in a regime where effective field theory is applicable.

Figures (16)

• Figure 1: Representitive plots for the evolution of $\rho$ (top left), $\theta$ (top right), $\theta^{(0,1)}$ (bottom left) and $\theta^{(0,2)}$ (bottom right). $\rho$ and $\theta$ are the solutions of the UV theory while $\theta^{(0,1)}$ and $\theta^{(0,2)}$ are solutions of the EFT$_4$. We use \ref{['eq:init_dat_1']} as initial data and set $M=100$ and $\alpha_1=-0.8$, $\alpha_2=0.3$ to evolve the EFT$_4$.
• Figure 2: The spectral density of the fields plotted in Fig. \ref{['fig:all_plots']}: $\rho$ (top left), $\theta$ (top right), $\theta^{(0,1)}$ (bottom left) and $\theta^{(0,2)}$ (bottom right).
• Figure 3: $C_t\,L^2_x$ norm of the difference between the UV solution and the various EFTs, for $M=100$ and $M=1000$. Note that the EFT$_1$ and EFT$_2$ provide equally accurate solutions for both choices of UV mass scale, and their respective curves on this plot are on top of each other.
• Figure 4: $C_t\,H^1_x$ norm of the difference between the EFT solutions and the UV solution remain small and bounded throughout the evolution.
• Figure 5: Effect of field redefinitions on the solution for the regularised EFTs. For the EFT$_2$, the effect of the field definition \ref{['eq:field_redef_M2']} is of higher order and does not affect the accuracy of the solution. On the other hand, for the EFT$_4$, the solution for $\tilde{\theta}$ (defined by \ref{['eq:field_redef_M4']}) provides a noticeably better approximation to the UV solution than the bare solution $\theta$.