Effective Field Theories as Asymptotic Series: From QCD to Cosmology

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.