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The simple scheme for the calculation of the anomalous dimensions of composite operators in the 1/N expansion

S. E. Derkachov, A. N. Manashov

Abstract

The simple method for the calculating of the anomalous dimensions of the composite operators up to 1/N^2 order is developed. We demonstrate the effectiveness of this approach by computing the critical exponents of the $(\otimes\vecΦ)^{s}$ and $\vecΦ\otimes(\otimes\vec\partial)^{n}\vecΦ$ operators in the 1/N^2 order in the nonlinear sigma model. The special simplifications due to the conformal invariance of the model are discussed.

The simple scheme for the calculation of the anomalous dimensions of composite operators in the 1/N expansion

Abstract

The simple method for the calculating of the anomalous dimensions of the composite operators up to 1/N^2 order is developed. We demonstrate the effectiveness of this approach by computing the critical exponents of the and operators in the 1/N^2 order in the nonlinear sigma model. The special simplifications due to the conformal invariance of the model are discussed.

Paper Structure

This paper contains 9 sections, 57 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The extended model
  4. Conformal structures in perturbation theory.
  5. The anomalous dimensions of the $(\otimes \vec{\Phi})^{s}$ and $(\vec{\Phi}\otimes(\otimes \vec{\partial})^{n} \vec{\Phi})$ operators in $1/N^{2}$ order
  6. Conclusion
  7. The renormalized Green functions in $1/N$ order
  8. Values for the $1/N^{2}$ order integrals
  9. Chain rule and uniqueness relation

Figures (10)

  • Figure 1: The example of the rearrangement of the diagrams. The black dots denote the vertex counterterm. The symbol ${\cal R}^{\prime}$ stands for the standard operation of the subtracting of the subdivergencies from a diagram.
  • Figure 2: The effective $\psi$ - line
  • Figure 3: The $1/N$ - diagrams for the Green function with the insertion of the operator $\phi_{A}\phi_{B}$.
  • Figure 4: The $1/N^2$ - diagrams for the Green function with insertion of operator $\phi_{A}\phi_{B}$.
  • Figure 5: The $1/N$ order vertex diagrams
  • ...and 5 more figures