Effective Field Theories as Asymptotic Series: From QCD to Cosmology

TL;DR

This paper argues that effective Lagrangians in physics are inherently asymptotic series with factorial divergence, impacting various fields from QCD to cosmology by highlighting the limitations of their convergence.

Contribution

It demonstrates the universal factorial divergence in effective field theories and explores its implications across multiple areas of physics, including QCD, gauge theories, lattice models, and cosmology.

Findings

Effective Lagrangians are asymptotic series with factorial divergence.

The factorial behavior appears in diverse physical contexts such as QCD and cosmology.

Implications of asymptotic series for understanding physical phenomena.

Abstract

We present some generic arguments demonstrating that an effective Lagrangian $L_{eff}$ which, by definition, contains operators $O^n$ of arbitrary dimensionality in general is not convergent, but rather an asymptotic series. It means that the behavior of the far distant terms has a specific factorial dependence $L_{eff}\sim \sum_n \frac{c_n O^n}{M^{n}},~c_n\sim n! ,~n\gg1$. We discuss a few apparently different problems, which however have something in common-- the aforementioned $n!-$ behavior: 1.Effective long -distance theory describing the collective fields in QCD; 2.Effective Berry phase potential which is obtained by integrating over the fast degrees of freedom. As is known, the Berry potential is associated with induced local gauge symmetry and might be relevant for the compactification problem at the Planck scale. 3.Nonlocal Lagrangians introduced by Georgi\cite{Georgi} for appropriate treatment of the effective field theories without power expanding. 4.The so-called improved action in lattice field theory where the new, higher dimensional operators have been introduced into the theory in order to reduce the lattice artifacts. 5.Cosmological constant problem and vacuum expectation values in gravity. We discuss some applications of this, seemingly pure academic phenomenon, to various physical problems with typical energies from $1 GeV$ to the Plank scale.