Abstract
We present an algorithm which allows to solve analytically linear systems of differential equations which factorize to first order. The solution is given in terms of iterated integrals over an alphabet where its structure is implied by the coefficient matrix of the differential equations. These systems appear in a large variety of higher order calculations in perturbative Quantum Field Theories. We apply this method to calculate the master integrals of the three-loop massive form factors for different currents, as an illustration, and present the results for the vector form factors in detail. Here the solution space emerging is given by the cyclotomic harmonic polylogarithms and their associated special constants. No special basis representation of the master integrals is needed. The algorithm can be applied as well to more general cases factorizing at first order, which are based on more general alphabets, iterated integrals and associated constants.
Highlights
The fundamental objects in any gauge theory are the scattering amplitudes or correlation functions, as they allow to compute the scattering cross sections for collider experiments at large facilities like the Large Hadron Collider (LHC) at CERN
We apply our algorithm to the single scale and first order factorizable system of differential equations which are relevant for the color-planar and the complete light quark contributions to the heavy quark form factors
We presented an algorithm to solve single-variate systems of differential equations, factorizing at first order and depending on the dimensional parameter ε, analytically
Summary
The fundamental objects in any gauge theory are the scattering amplitudes or correlation functions, as they allow to compute the scattering cross sections for collider experiments at large facilities like the Large Hadron Collider (LHC) at CERN. In the case where these systems factorize at first order, the complete solution can be constructed algorithmically This has been done before in Ref. We employ this method of integration for computing the set of MIs which contribute to both the color-planar and complete light quark non-singlet three-loop contributions to the heavy-quark form factors for different currents, namely the vector, axial-vector, scalar and pseudo-scalar currents. Using the method described in this paper, we have obtained both the colorplanar and complete light quark contributions to the three-loop form factors for the other three currents, namely axial-vector, scalar and pseudo-scalar currents in [52]. The complete expressions for the vector form factors, which are very large, are given in ancillary files together with the code CPOLY.f for the cyclotomic harmonic polylogarithms and other material, attached to this paper
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