The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements

Abstract

Future high luminosity polarized deep-inelastic scattering experiments will improve both the knowledge of the spin sub-structure of the nucleons and contribute further to the precision determination of the strong coupling constant, as well as, reveal currently yet unknown higher twist contributions in the polarized sector. For all these tasks to be performed, it is necessary to know the QCD leading twist scaling violations of the measured structure functions. Here an important ingredient consists in the polarized singlet anomalous dimensions and splitting functions in QCD. We recalculate these quantities to three-loop order in the M-scheme by using the traditional method of space-like off-shell massless operator matrix elements, being a gauge-dependent framework. Here one obtains the anomalous dimensions without referring to gravitational currents, needed when calculating them using the forward Compton amplitude. We also calculate the non-singlet splitting function $$ \Delta {P}_{\mathrm{qq}}^{(2),\mathrm{s},\mathrm{NS}} $$ ∆ P qq 2 , s , NS and compare the final results to the literature, also including predictions for the region of small values of Bjorken x.

The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements

“…The anomalous dimensions are iteratively extracted form the 1/ε pole terms and the other expansion coefficients a NS,(i,j) qq are given in Ref. [36]. Eq.…”

“…The present calculation requires the knowledge of the corresponding massless off shell OMEs to two-loop order, cf. [32,33,36], up to the terms of O(ε 0 ) in the dimensional parameter ε = D − 4. The off shell OMEs are gauge-dependent quantities.…”

The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements

“…The anomalous dimensions are iteratively extracted from the 1/ε pole terms and the other expansion coefficients a (k,l) ij are given in Ref. [48]. The anomalous dimensions in the M-scheme [19,20] 4 are obtained by the following transformations ∆γ…”

“…Ref. [48]. We use the integration-by-parts relations [54,72] to reduce the problem to master integrals.…”

“…The total number of irreducible diagrams in the polarized singlet case at three-loop order is larger by a factor of ∼ 22 than in the two-loop order. The reducible diagrams are accounted for by wave-function renormalization [44,46,47], decorating the OMEs at lower order in the coupling constant [19,48].…”