Towards three-loop QCD corrections to the time-like splitting functions

We report on the status of a direct computation of the time-like splitting functions at next-to-next-to-leading order in QCD. Time-like splitting functions govern the collinear kinematics of inclusive hadron production and the evolution of the parton fragmentation distributions. Current knowledge about them at three loops has been inferred by means of crossing symmetry from their related space-like counterparts, which has left certain parts of the off-diagonal quark-gluon splitting function undetermined. This motivates an independent calculation from first principles. We review the tools and methods which are applied to attack the problem.Comment: 11 pages, 5 figures; presented at the Epiphany Conference 2015 (Cracow, Poland); additional files: MassFactorization.n

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form

We present Fuchsia — an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients $∂_xJ(x,ε)=\mathbb{A}(x,ε)J(x,ε)$ finds a basis transformation $\mathbb{T}(x,ε),i.e., J(x,ε)= \mathbb{T}(x,ε)J′(x,ε)$, such that the system turns into the epsilon form: $∂_x J′(x,ε)=ε \mathbb{S}(x)J′(x,ε)$, where $\mathbb{S}(x)$, where is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator. That makes the construction of the transformation $\mathbb{T}(x,ε)$ crucial for obtaining solutions of the initial system.In principle, Fuchsia can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals

Fuchsia : A tool for reducing differential equations for Feynman master integrals to epsilon form

We present Fuchsia — an implementation of the Lee algorithm, which for a given system of ordinary differential equations with rational coefficients $∂_xJ(x,ε)=\mathbb{A}(x,ε)J(x,ε)$ finds a basis transformation $\mathbb{T}(x,ε),i.e., J(x,ε)= \mathbb{T}(x,ε)J′(x,ε)$, such that the system turns into the epsilon form: $∂_x J′(x,ε)=ε \mathbb{S}(x)J′(x,ε)$, where $\mathbb{S}(x)$, where is a Fuchsian matrix. A system of this form can be trivially solved in terms of polylogarithms as a Laurent series in the dimensional regulator. That makes the construction of the transformation $\mathbb{T}(x,ε)$ crucial for obtaining solutions of the initial system.In principle, Fuchsia can deal with any regular systems, however its primary task is to reduce differential equations for Feynman master integrals. It ensures that solutions contain only regular singularities due to the properties of Feynman integrals