Peculiarities of beta functions in sigma models

Abstract

In this paper we consider perturbation theory in generic two-dimensional sigma models in the so-called first-order formalism, using the coordinate regularization approach. Our goal is to analyze the first-order formalism in application to $\beta$ functions and compare its results with the standard geometric calculations. Already in the second loop, we observe deviations from the geometric results that cannot be explained by the regularization/renormalization scheme choices. Moreover, in certain cases the first-order calculations produce results that are not symmetric under the classical diffeomorphisms of the target space. Although we could not present the full solution to this remarkable phenomenon, we found some indirect arguments indicating that an anomaly similar to that established in supersymmetric Yang-Mills theory might manifest itself starting from the second loop. We discuss why the difference between two answers might be an infrared effect, similar to that in $\beta$ functions in supersymmetric Yang-Mills theories. In addition to the generic Kähler target spaces we discuss in detail the so-called Lie-algebraic sigma models. In particular, this is the case when the perturbed field $G^{i\bar{j}}$ is a product of the holomorphic and antiholomorphic currents satisfying two-dimensional current algebra.

arXiv comment31 pages, 2 figures

References in corpus (21)

• Integrable interpolations: From exact CFTs to non-Abelian T-duals
• Holomorphy, Rescaling Anomalies and Exact beta Functions in Supersymmetric Gauge Theories
• One-Dimensional Quasi-Exactly Solvable Schrödinger Equations
• The all-loop non-Abelian Thirring model and its RG flow
• Derivation of the exact NSVZ beta-function in N=1 SQED, regularized by higher derivatives, by direct summation of Feynman diagrams
• On sigma model RG flow, "central charge" action and Perelman's entropy
• Gauged WZW-type theories and the all-loop anisotropic non-Abelian Thirring model
• On the Beta Function for Anisotropic Current Interactions in 2D
• On First Order Formalism in String Theory
• All-loop correlators of integrable $\lambda$-deformed $\sigma$-models
• The 4-loop beta-function in the 2D Non-Abelian Thirring model, and comparison with its conjectured "exact" form
• The all-loop perturbative derivation of the NSVZ $\beta$-function and the NSVZ scheme in the non-Abelian case by summing singular contributions
• Flag manifold sigma models: spin chains and integrable theories
• N=(0,2) Deformation of CP(1) Model: Two-dimensional Analog of N=1 Yang-Mills Theory in Four Dimensions
• The NSVZ $\beta$-function for theories regularized by higher covariant derivatives: the all-loop sum of matter and ghost singularities
• Perturbed Beta-Gamma Systems and Complex Geometry
• BRST, Generalized Maurer-Cartan Equations and CFT
• RG flows for $\lambda$-deformed CFTs
• Beta-gamma systems and the deformations of the BRST operator
• First Order String Theory and the Kodaira-Spencer Equations. I
• First Order String Theory and the Kodaira-Spencer Equations. II

Cited by in corpus (3)

• Supersymmetric Grassmannian Sigma Models in Gross-Neveu Formalism
• Higher-Dimensional Chirally Stabilized Fixed Points and Their Deformations
• Nonperturbative features in the Lie-algebraic Kähler sigma model with fermions