Abstract
We consider renormalization of four-fermion operators in the critical QED and $SU(N_c)$ version of Gross--Neveu--Yukawa model in non-integer dimensions. Since the number of mixing operators is infinite, the diagonalization of an anomalous dimension matrix becomes a nontrivial problem. At leading order, construction of eigen-operators is equivalent to solving certain three-term recurrence relations. We find analytic solutions of these recurrence relations that allows to determine the spectrum of anomalous dimensions and study their properties.
Highlights
Quantum field theories (QFTs) in noninteger dimensions d < 4 were introduced as a tool to calculate critical exponents in three-dimensional systems at a phase transition point [1]
We consider the renormalization of four-fermion operators in the critical QED and SUðNcÞ version of the Gross-Neveu-Yukawa model in noninteger dimensions
We find analytic solutions of these recurrence relations that allow us to determine the spectrum of anomalous dimensions and study their properties
Summary
Quantum field theories (QFTs) in noninteger dimensions d < 4 were introduced as a tool to calculate critical exponents in three-dimensional systems at a phase transition point [1]. As a rule, QFTs in d 1⁄4 4 − 2ε possess nontrivial critical points with coupling constants being of order ε. 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. Γnμ1...μn is the antisymmetrized product of the d-dimensional γ-matrices All these operators have canonical dimension Δ 1⁄4 6 and mix under renormalization. P bet∞nw1⁄4e0ecnn two eigenoperators ðΔÞOn, to be finite, i.e., for OΔ hOΔðxÞOΔð0Þi < ∞: ð2Þ This condition is always fulfilled if the mixing matrix has a finite size, as in the case of scalar field theories, but leads to nontrivial “quantization” conditions for infinite matrices. IV and study the renormalization of four-fermion operators in this model
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