Calculating massive 3-loop graphs for operator matrix elements by the method of hyperlogarithms

Jakob Ablinger,Johannes Blümlein,Clemens Raab,Carsten Schneider,Fabian Wißbrock

doi:10.1016/j.nuclphysb.2014.04.007

Jakob Ablinger, Johannes Blümlein + Show 3 more

Open Access

https://doi.org/10.1016/j.nuclphysb.2014.04.007

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Abstract

We calculate convergent 3-loop Feynman diagrams containing a single massive loop equipped with twist τ=2 local operator insertions corresponding to spin N. They contribute to the massive operator matrix elements in QCD describing the massive Wilson coefficients for deep-inelastic scattering at large virtualities. Diagrams of this kind can be computed using an extended version of the method of hyperlogarithms, originally being designed for massless Feynman diagrams without operators. The method is applied to Benz- and V-type graphs, belonging to the genuine 3-loop topologies. In case of the V-type graphs with five massive propagators, new types of nested sums and iterated integrals emerge. The sums are given in terms of finite binomially and inverse binomially weighted generalized cyclotomic sums, while the 1-dimensionally iterated integrals are based on a set of ∼30 square-root valued letters. We also derive the asymptotic representations of the nested sums and present the solution for N∈C. Integrals with a power-like divergence in N-space ∝aN,a∈R,a>1, for large values of N emerge. They still possess a representation in x-space, which is given in terms of root-valued iterated integrals in the present case. The method of hyperlogarithms is also used to calculate higher moments for crossed box graphs with different operator insertions.

Highlights

Massive on-shell operator matrix elements (OMEs) occur in the calculation of the Wilson coefficients in deeply-inelastic scattering, describing these quantities at large enough virtualitiesQ2 m2 together with the massless Wilson coefficients [1]
That far we have described the algorithm for a finite loop diagram built of propagators and vertices for a renormalizable quantum field theory
A direct method was sought for to arrive at these results right form the Feynman parameterization of the contributing diagrams

Summary

Introduction

Massive on-shell operator matrix elements (OMEs) occur in the calculation of the Wilson coefficients in deeply-inelastic scattering, describing these quantities at large enough virtualities. [41] we calculated diagrams of the 3-loop ladder topology of up to six massive propagators, including the most demanding cases. Not all of these graphs could be calculated using the above technologies. In case the corresponding graph exhibits no poles in the dimensional parameter ε = D − 4, the method of hyperlogarithms has been devised for massless 2-point topologies with an off-shell external momentum in scalar field theory in Ref. They may be considered to emerge from either a ladder- or the crossed box-topology by removing one line While in the former case conventional structures are obtained, in the latter case new nested sum-types emerge, which contain weights due to binomials of the type.

The Formalism

Benz-Graphs

V-type Diagrams with Five Massive Propagators

Analytic Continuation of Binomially Weighted Nested Sums

Moments for Crossed-Box Graphs

Conclusions