We investigate the single off-shell scalar box integral with massless internal lines in dimensional regularization. A special emphasis is given to higher orders in the dimensional regularization parameter epsilon, its branch cut structure, and kinematic limits. Common representations of the box integral introduce superficial branch cuts, which we eliminate to all orders in the epsilon expansion. In the literature so far a satisfactory solution for this issue only exists up to finite order in the epsilon expansion. However, for certain calculations at NNLO in perturbation theory, higher orders in epsilon of this integral are required. In this paper, we present results to all orders in epsilon in terms of single-valued polylogarithms and explicitly determine the real and imaginary part of the box integral in all kinematic regions.

We discuss new ideas for consideration of loop diagrams and angular integrals in D-dimensions in QCD. In case of loop diagrams, we propose the covariant formalism of expansion of tensorial loop integrals into the orthogonal basis of linear combinations of external momenta. It gives a very simple representation for the final results and is more convenient for calculations on computer algebra systems. In case of angular integrals we demonstrate how to simplify the integration of differential cross sections over polar angles. Also we derive the recursion relations, which allow to reduce all occurring angular integrals to a short set of basic scalar integrals. All order ε-expansion is given for all angular integrals with up to two denominators based on the expansion of the basic integrals and using recursion relations. A geometric picture for partial fractioning is developed which provides a new rotational invariant algorithm to reduce the number of denominators.

We generalize the result of our recent paper on the massless single off-shell scalar box integral to the case of two non-adjacent end points off the light cone. An analytic result in d = 4 − 2ε dimensions is established in terms of four Gauss hypergeometric 2 F 1 functions respectively their single-valued counterparts. This allows for an explicit splitting of real and imaginary parts, as well as an all-order ε-expansion in terms of single-valued polylogarithms.

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We generalize the result of our recent paper on the massless single off-shell scalar box integral to the case of two non-adjacent end points off the light cone. An analytic result in d = 4 − 2ε dimensions is established in terms of four Gauss hypergeometric 2F1 functions respectively their single-valued counterparts. This allows for an explicit splitting of real and imaginary parts, as well as an all-order ε-expansion in terms of single-valued polylogarithms.