Renormalization scheme and gauge (in)dependence of the generalized Crewther relation: what are the real grounds of the β-factorization property?

• Published in: The journal of high energy physics Vol. 2018; no. 2; pp. 1 - 46
• Main Authors: Garkusha, A. V., Kataev, A. L., Molokoedov, V. S.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2018; Springer Nature B.V; SpringerOpen
• Subjects: Classical and Quantum Gravitation; Elementary Particles; Factorization; Flavor (particle physics); High energy physics; Invariants; Perturbation theory; Perturbative QCD; Physics; Physics and Astronomy; Quantum Field Theories; Quantum Field Theory; Quantum Physics; Regular Article - Theoretical Physics; Relativity Theory; Renormalization Group; String Theory; Studies; Subtraction; Renormalization Group; Perturbative QCD
• ISSN: 1029-8479; 1029-8479
• DOI: 10.1007/JHEP02(2018)161

A bstract The problem of scheme and gauge dependence of the factorization property of the renormalization group β -function in the SU( N c ) QCD generalized Crewther relation (GCR), which connects the flavor non-singlet contributions to the Adler and Bjorken polarized sum rule functions, is investigated at the O a s 4 level of perturbation theory. It is known that in the gauge-invariant renormalization M S ¯ -scheme this property holds in the QCD GCR at least at this order. To study whether this factorization property is true in all gauge-invariant schemes, we consider the MS-like schemes in QCD and the QED-limit of the GCR in the M S ¯ -scheme and in two other gauge-independent subtraction schemes, namely in the momentum MOM and the on-shell OS schemes. In these schemes we confirm the existence of the β -function factorization in the QCD and QED variants of the GCR. The problem of the possible β -factorization in the gauge-dependent renormalization schemes in QCD is studied. To investigate this problem we consider the gauge non-invariant mMOM and MOMgggg-schemes. We demonstrate that in the mMOM scheme at the O a s 3 level the β -factorization is valid for three values of the gauge parameter ξ only, namely for ξ = −3 , −1 and ξ = 0. In the O a s 4 order of PT it remains valid only for case of the Landau gauge ξ = 0. The consideration of these two gauge-dependent schemes for the QCD GCR allows us to conclude that the factorization of RG β -function will always be implemented in any MOM-like renormalization schemes with linear covariant gauge at ξ = 0 and ξ = −3 at the O a s 3 approximation. It is demonstrated that if factorization property for the MS-like schemes is true in all orders of PT, as theoretically indicated in the several works on the subject, then the factorization will also occur in the arbitrary MOM-like scheme in the Landau gauge in all orders of perturbation theory as well.