^{1}

^{,*}

^{2,1,3}

^{,†}

^{3}.

We consider the renormalization of four-fermion operators in the critical QED and

Quantum field theories (QFTs) in noninteger dimensions

It is clear, however, that QFTs in noninteger dimensions are not full fledged quantum field models—no real physical system is described by these QFTs. Thus they are not obliged to comply with expectations based on physical principles. It was shown in Ref.

Physical observables in conformal field theories (CFTs) are correlation functions of local operators. One is interested, in particular, in their behavior under scale and conformal transformations. Therefore the basis of operators which transform in a proper way under scale and conformal transformations plays a distinguished role. In perturbation theory, such a basis is constructed by diagonalization of the anomalous dimension matrices. Since only operators of the same canonical dimension mix under renormalization, such a matrix has a finite size in scalar field theories. In a fermionic QFT, the situation is quite different—the number of mixing operators is, in most cases, infinite. The simplest example of this kind is given by the four fermion operators,

In the QCD context, four-fermion operators arise in the description of nonleptonic weak decays of hadrons. Their renormalization was studied in

Here we are interested in a different challenge—constructing operators which have certain scaling dimensions at a critical point. Since the size of the mixing matrix for the operators

In this work, we consider renormalization of four-fermion operators in two theories: the critical QED and

The paper is organized as follows: In Sec.

In

The model can also be analyzed with the

Renormalization of four-fermion operators

The transpose matrix

All other eigenvectors of the matrix

The leading order correlator of two basic operators (all fields have different flavors) was calculated in

Then for the eigenoperators

The relation inverse to Eq.

Coming back to the physical operators

Since the operators

In this section, we briefly consider the specifics of operator mixing in the GNY model

Surprisingly enough, the matrix

The explanation for such degeneracy of the anomalous dimensions is the following: let us consider two sets of operators,

In a similar manner, one can easily show that the vector

Thus, we conclude that as long as the matrix

We consider the

Numerical analysis shows that the stable critical point exists for all

Let us study the renormalization of four-fermion operators in this model. First, we note that the operators

At the critical point, the RGE for the operators

For

For

As was discussed in Sec.

For the “physical” operators (such that not all

All other solutions of the recurrence relation

In order to fix the allowed values of

One notices that there is a certain resemblance between the anomalous dimensions of four-fermion operators in the

In QED, the anomalous dimension matrix in the physical sector at two loops was obtained in

Finally, we consider an example to show that the construction of operators with “good” scaling properties is not always possible. Let

Proceeding along the same lines with the correlator

This statement can also be formulated as follows. The matrix

The conclusion is that in noninteger dimensions, the possibility of representing the correlator

We have considered the renormalization of four-fermion operators in the critical QED and extended GNY models. The anomalous dimension matrix in both models is of infinite size so that in order to make the diagonalization problem well defined, additional restrictions have to be imposed on the solutions. It is natural to demand for the correlation functions of the eigenoperators to be finite in the

It is expected that in the

We are grateful to Michael Kelly for collaboration in the early stages of this project and V. Braun for useful comments. This work was supported by the DFG Grants No. BR 2021/7-1 (Y. J.), No. MO 1801/1-3 (A. M.) and by Russian Scientific Foundation Project No. 14-11-00598 (A. M.).

The antisymmetrized product of

Here we collected some basic facts about the dual continuous Hahn polynomials,