Abstract
We introduce a class of iterated integrals, defined through a set of linearly independent integration kernels on elliptic curves. As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring that they have at most simple poles, implying that the iterated integrals have at most logarithmic singularities. We study the properties of our iterated integrals and their relationship to the multiple elliptic polylogarithms from the mathematics literature. On the one hand, we find that our iterated integrals span essentially the same space of functions as the multiple elliptic polylogarithms. On the other, our formulation allows for a more direct use to solve a large variety of problems in high-energy physics. We demonstrate the use of our functions in the evaluation of the Laurent expansion of some hypergeometric functions for values of the indices close to half integers.
Highlights
The discovery of the Higgs boson at the LHC and the absence of clear signs of New Physics close to the electroweak scale provide a strong confirmation of the Standard Model of particles physics (SM) up to the TeV scale
As a direct generalisation of multiple polylogarithms, we construct our set of integration kernels ensuring that they have at most simple poles, implying that the iterated integrals have at most logarithmic singularities
In this paper we have introduced a class of iterated integrals on elliptic curves that have at most logarithmic singularities, and deserve to be called elliptic polylogarithms
Summary
The discovery of the Higgs boson at the LHC and the absence of clear signs of New Physics close to the electroweak scale provide a strong confirmation of the Standard Model of particles physics (SM) up to the TeV scale. The appearance of non-polylogarithmic structures in quantum field theory had been noticed already more than fifty years ago in the computation of the two-loop corrections to the electron self-energy in QED [32] The origin of this new class of functions could be traced back to a particular Feynman graph, the so-called two-loop massive sunrise graph, whose analytical properties have been extensively analyzed from different perspectives in the physics and mathematics literature [15,16,17,18,19,20,21,22,23, 33,34,35,36,37,38].1. In appendix A we discuss some technical aspects about regularisation of iterated integrals omitted in the main text
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