English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The unpolarized and polarized massive operator matrix elements $A_{Qg}^{(3)}$and $\Delta A_{Qg}^{(3)}$ contain first-order factorizable and non-first-orderfactorizable contributions in the determining difference or differentialequations of their master integrals. We compute their first-order factorizablecontributions in the single heavy mass case for all contributing Feynmandiagrams. Moreover, we present the complete color-$\zeta$ factors for the casesin which also non-first-order factorizable contributions emerge in the masterintegrals, but cancel in the final result as found by using the method ofarbitrary high Mellin moments. Individual contributions depend also ongeneralized harmonic sums and on nested finite binomial and inverse binomialsums in Mellin $N$-space, and correspondingly, on Kummer-Poincar\'e andsquare-root valued alphabets in Bjorken-$x$ space. We present a completediscussion of the possibilities of solving the present problem in $N$-spaceanalytically and we also discuss the limitations in the present case toanalytically continue the given $N$-space expressions to $N \in \mathbb{C}$ bystrict methods. The representation through generating functions allows a wellsynchronized representation of the first-order factorizable results over a17-letter alphabet. We finally obtain representations in terms of iteratedintegrals over the corresponding alphabet in $x$-space, also containing up toweight {\sf w = 5} special constants, which can be rationalized toKummer-Poincar\'e iterated integrals at special arguments. The analytic$x$-space representation requires separate analyses for the intervals $x \in[0,1/4], [1/4,1/2], [1/2,1]$ and $x > 1$. We also derive the small and large$x$ limits of the first-order factorizable contributions.

The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$