Abstract
Conformal symmetry-based relations between concrete perturbative QED and QCD approximations for the Bjorken , the Ellis-Jaffe sum rules of polarized lepton- nucleon deep-inelastic scattering (DIS), the Gross-Llewellyn Smith sum rules of neutrino-nucleon DIS, and for the Adler functions of axial-vector and vector channels are derived. They result from the application of the operator product expansion to three triangle Green functions, constructed from the non-singlet axial-vector, and two vector currents, the singlet axial-vector and two non-singlet vector currents and the non-singlet axial-vector, vector and singlet vector currents in the limit, when the conformal symmetry of the gauge models with fermions is considered unbroken. We specify the perturbative conditions for this symmetry to be valid in the case of the U(1) and SU(N c) models. The all-order perturbative identity following from the conformal invariant limit between the concrete contributions to the Bjorken, the Ellis-Jaffe and the Gross-Llewellyn Smith sum rules is proved. The analytical and numerical O(α 4) and $ O\left( {\alpha_s^2} \right) $ conformal symmetry based approximations for these sum rules and for the Adler function of the non-singlet vector currents are summarized. Possible theoretical applications of the results presented are discussed.
Highlights
Formulated in more detail in [11], the cancellation of all 2-loop internal contributions to the AVV three-point function were rediscovered in ref. [12]
Conformal symmetry-based relations between concrete perturbative QED and QCD approximations for the Bjorken, the Ellis-Jaffe sum rules of polarized lepton- nucleon deep-inelastic scattering (DIS), the Gross-Llewellyn Smith sum rules of neutrino-nucleon DIS, and for the Adler functions of axial-vector and vector channels are derived. They result from the application of the operator product expansion to three triangle Green functions, constructed from the non-singlet axial-vector, and two vector currents, the singlet axialvector and two non-singlet vector currents and the non-singlet axial-vector, vector and singlet vector currents in the limit, when the conformal symmetry of the gauge models with fermions is considered unbroken
Where Aμ(y) = ψ(y)γμγ5ψ(y) is the SI axial-vector fermion current. The properties of this Green function were investigated previously in ref. [14] within the deeply investigated finite QED program. This program had the aim to find out whether a nontrivial ultraviolet zero may exist in the RG β-function of the perturbative quenched QED model or in the QED Gell-Mann-Low function Ψ(α) [17], which as clarified in the review of ref. [18] is identical to the QED β-function in the momentum subtractions scheme
Summary
Let us study the three-point functions of eq (1.2) and eq (1.3) in the conformally invariant limits of the U(1) and SU (Nc) gauge models with fermions. As will be discussed below, in the Abelian U(1) model with fermions, these requirements can be formulated in diagrammatic language and are described by the blocks of Feynman graphs, where the QED coupling constant a = α/π is fixed and is not renormalized. This leads to the property Z3 = 1, where Z3 is the renormalization constant of the photon propagator, which is related to the renormalization of the QED coupling constant by a = Z3aB, where a = α/π. In the CS limit the application of the operator product expansion (OPE) approach to eq (1.1), eq (2.3) and eq (2.4) allow us to derive relations between the approximations for the coefficient functions of the DIS sum rules, which will be defined below
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