Abstract
We present an algorithm of the reduction of the differential equations for master integrals the Fuchsian form with the right-hand side matrix linearly depending on dimensional regularization parameter $\epsilon$. We consider linear transformations of the functions column which are rational in the variable and in $\epsilon$. Apart from some degenerate cases described below, the algorithm allows one to obtain the required transformation or to ascertain irreducibility to the form required. Degenerate cases are quite anticipated and likely to correspond to irreducible systems.
Highlights
Used for determining whether a specific integral is homogeneous or not, in general, they do not give an algorithm of finding appropriate basis
We present an algorithm of the reduction of the differential equations for master integrals the Fuchsian form with the right-hand side matrix linearly depending on dimensional regularization parameter
In the present paper we describe a method of finding an appropriate basis which is based on the differential system alone
Summary
We consider the system of differential equations for the master integrals as given in eq (1.1). The differential system (1.1) is said to have Poincare rank p 0 at the singular point x = x0 = ∞ if M(x) can be represented as M(x) = A(x − x0)/(x − x0)1+p, where A(x) is regular at x = x0 matrix and A(0) = 0. If all singularities are regular, after the application of the algorithm, Poincare ranks for all but one singularities can be nullified and the system is reduced to a Fuchsian form everywhere, except, may be, one point. The possibility to transform a regular system to Fuchsian form in all points and to eliminate all apparent singularities would mean the positive solution of the 21st Hilbert problem, consisting of proving of the existence of linear differential equations having a prescribed monodromy group.
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