[Paper Review] Effective Field Theories as Asymptotic Series: From QCD to Cosmology

Ariel Zhitnitsky|arXiv (Cornell University)|Jan 26, 1996

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper demonstrates that effective field theories (EFTs) in quantum chromodynamics (QCD) and cosmology are generically asymptotic series, with coefficients growing factorially as $ c_n \sim n! $, leading to divergence at high orders. The author shows this factorial behavior underlies diverse phenomena—from low-energy QCD dynamics and Berry phase potentials to improved lattice actions and the cosmological constant problem—revealing a deep, universal structure in EFTs across energy scales from 1 GeV to the Planck scale.

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.

Motivation & Objective

• To establish that effective field theories (EFTs) with higher-dimensional operators are generically asymptotic series rather than convergent expansions.
• To identify the universal $ n! $ factorial growth of coefficients in EFTs as a common feature across seemingly unrelated physical systems.
• To connect this asymptotic behavior to physical phenomena such as the Berry phase potential, improved lattice actions, and the cosmological constant problem.
• To demonstrate that this divergence is not a flaw but a fundamental property of EFTs, with implications for quantum gravity and compactification at the Planck scale.

Proposed method

• Analyzes the structure of effective Lagrangians $ L_{\text{eff}} \sim \sum_n \frac{c_n O^n}{M^n} $, where $ c_n \sim n! $ for large $ n $, indicating asymptotic behavior.
• Applies the method of resummation and Borel resummation techniques to analyze divergent series in EFTs.
• Examines the role of integrating out fast degrees of freedom, leading to non-perturbative potentials such as the Berry phase potential.
• Considers the introduction of higher-dimensional operators in lattice field theory (improved actions) as a mechanism to suppress lattice artifacts, tied to asymptotic series behavior.
• Draws parallels between the factorial divergence in EFTs and the behavior of vacuum expectation values in gravity and the cosmological constant problem.
• Uses dimensional regularization and effective field theory techniques to study the scaling of operators and their coefficients across energy scales.

Experimental results

Research questions

• RQ1 Why do effective field theories in QCD and gravity exhibit divergent series with factorial growth in their coefficients?
• RQ2 How is the $ n! $-behavior in EFTs connected to the emergence of the Berry phase potential in low-energy dynamics?
• RQ3 What is the role of higher-dimensional operators in lattice field theory, and how does their inclusion relate to asymptotic series?
• RQ4 Can the cosmological constant problem be understood through the lens of divergent EFT expansions with factorial coefficients?
• RQ5 Is the asymptotic nature of EFTs a universal feature across quantum field theories, from QCD to quantum gravity?

Key findings

• The effective Lagrangian in EFTs is generically an asymptotic series with coefficients growing as $ c_n \sim n! $, leading to divergence at high orders.
• This factorial behavior unifies diverse phenomena: low-energy QCD dynamics, Berry phase potentials from adiabatic transport, and improved lattice actions with higher-dimensional operators.
• The Berry phase potential, arising from integrating out fast degrees of freedom, exhibits the same $ n! $-dependence and is linked to induced local gauge symmetry and Planck-scale compactification.
• The cosmological constant problem is shown to be related to the same asymptotic structure, suggesting a deep connection between vacuum energy and divergent EFT expansions.
• Improved actions in lattice field theory, which include higher-dimensional operators to reduce artifacts, are naturally described by such asymptotic series.
• The paper establishes that the divergence is not an artifact but a fundamental feature of EFTs, with implications for quantum gravity and the structure of spacetime at high energies.