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Jakob Ablinger, Arnd Behring, J. Blümlein, A. Freitas, Andreas von Manteuffel, Carsten Schneider, Kay Schönwald
Abstract
The non-first-order-factorizable contributions 1 to the unpolarized and polarized massive operator matrix elements to three-loop order, A Q g ( 3 ) and Δ A Q g ( 3 ) , are calculated in the single-mass case. For the F 1 2 -related master integrals of the problem, we use a semi-analytic method based on series expansions and utilize the first-order differential equations for the master integrals which does not need a special basis of the master integrals. Due to the singularity structure of this basis a part of the integrals has to be computed to O ( ε 5 ) in the dimensional parameter. The solutions have to be matched at a series of thresholds and pseudo-thresholds in the region of the Bjorken variable x ∈ ] 0 , ∞ [ using highly precise series expansions to obtain the imaginary part of the physical amplitude for x ∈ ] 0 , 1 ] at a high relative accuracy. We compare the present results both with previous analytic results, the results for fixed Mellin moments, and a prediction in the small- x region. We also derive expansions in the region of small and large values of x . With this paper, all three-loop single-mass unpolarized and polarized operator matrix elements are calculated.