A 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 (⊗ Φ) s and ( Φ⊗(⊗ ∂) n Φ) 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.

Correction exponents in the Gross–Neveu–Yukawa model at $$1/N^2$$ 1 / N 2

“…Our results are in complete agreement with the expressions for the perturbative four loop beta functions in the GNY model obtained recently in [19]. For the calculation we used the method developed in [21][22][23]. This method allows one to use the standard RG technique and, at the same time, to resum effectively a certain subset of diagrams reducing the total number of diagrams to be calculated.…”

“…Due to the non-multiplicative character of the renormalization the anomalous dimensions of the fields and operators cannot be, in general, determined via the corresponding renormalization factors. However, as it was shown in [23] this problem appears starting from 1/n 3 order only. Up to the order 1/n 2 the anomalous dimension of the operators can be obtained via the renormalization factors as follows.…”

“…-Instead of calculating separately each diagram of the second group, that is not quite simple, we use the result of [23] where it was shown that the contribution of all such diagrams to the anomalous dimension can be extracted from the 1/n order diagrams with dressed propagators and vertices. We discuss details of the calculation in Appendix B.…”

“…It was shown in [23] that the contribution to an anomalous dimension coming from diagrams with self-energy insertions and vertex corrections to a LO diagram can be obtained by evaluation the LO diagram with dressed propagators and vertices. The dressed propagators have the form 3…”

“…These methods are very effective for the calculation of the critical indices of the basic and auxiliary fields, but are not very suitable in the case of local operators. Therefore we use a different method developed in [21][22][23]. A detailed description of the method can be found in [23,24] and an application to the GN model in [25].…”