34 research outputs found
Independence of hyperlogarithms over function fields via algebraic combinatorics
We obtain a necessary and sufficient condition for the linear independence of
solutions of differential equations for hyperlogarithms. The key fact is that
the multiplier (i.e. the factor in the differential equation ) has
only singularities of first order (Fuchsian-type equations) and this implies
that they freely span a space which contains no primitive. We give direct
applications where we extend the property of linear independence to the largest
known ring of coefficients
Nested (inverse) binomial sums and new iterated integrals for massive Feynman diagrams
Nested sums containing binomial coefficients occur in the computation of
massive operator matrix elements. Their associated iterated integrals lead to
alphabets including radicals, for which we determined a suitable basis. We
discuss algorithms for converting between sum and integral representations,
mainly relying on the Mellin transform. To aid the conversion we worked out
dedicated rewrite rules, based on which also some general patterns emerging in
the process can be obtained.Comment: 13 pages LATEX, one style file, Proceedings of Loops and Legs in
Quantum Field Theory -- LL2014,27 April 2014 -- 02 May 2014 Weimar, German
Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums
The construction of Mellin-Barnes (MB) representations for non-planar Feynman
diagrams and the summation of multiple series derived from general MB
representations are discussed. A basic version of a new package AMBREv.3.0 is
supplemented. The ultimate goal of this project is the automatic evaluation of
MB representations for multiloop scalar and tensor Feynman integrals through
infinite sums, preferably with analytic solutions. We shortly describe a
strategy of further algebraic summation.Comment: Contribution to the proceedings of the Loops and Legs 2014 conferenc
On the solutions of universal differential equation by noncommutative Picard-Vessiot theory
Basing on Picard-Vessiot theory of noncommutative differential equations and
algebraic combinatorics on noncommutative formal series with holomorphic
coefficients, various recursive constructions of sequences of grouplike series
converging to solutions of universal differential equation are proposed. Basing
on monoidal factorizations, these constructions intensively use diagonal series
and various pairs of bases in duality, in concatenation-shuffle bialgebra and
in a Loday's generalized bialgebra. As applications, the unique solution,
satisfying asymptotic conditions, of Knizhnik-Zamolodchikov equations is
provided by d\'evissage
A localized version of the basic triangle theorem
In this short note, we give a localized version of the basic triangle theorem, first published in 2011 (see [4]) in order to prove the independence of hyperlogarithms over various function fields. This version provides direct access to rings of scalars and avoids the recourse to fraction fields as that of meromorphic functions for instance
Feynman integrals and hyperlogarithms
We study Feynman integrals in the representation with Schwinger parameters
and derive recursive integral formulas for massless 3- and 4-point functions.
Properties of analytic (including dimensional) regularization are summarized
and we prove that in the Euclidean region, each Feynman integral can be written
as a linear combination of convergent Feynman integrals. This means that one
can choose a basis of convergent master integrals and need not evaluate any
divergent Feynman graph directly.
Secondly we give a self-contained account of hyperlogarithms and explain in
detail the algorithms needed for their application to the evaluation of
multivariate integrals. We define a new method to track singularities of such
integrals and present a computer program that implements the integration
method.
As our main result, we prove the existence of infinite families of massless
3- and 4-point graphs (including the ladder box graphs with arbitrary loop
number and their minors) whose Feynman integrals can be expressed in terms of
multiple polylogarithms, to all orders in the epsilon-expansion. These
integrals can be computed effectively with the presented program.
We include interesting examples of explicit results for Feynman integrals
with up to 6 loops. In particular we present the first exactly computed
counterterm in massless phi^4 theory which is not a multiple zeta value, but a
linear combination of multiple polylogarithms at primitive sixth roots of unity
(and divided by ). To this end we derive a parity result on the
reducibility of the real- and imaginary parts of such numbers into products and
terms of lower depth.Comment: PhD thesis, 220 pages, many figure
On the algebraic structure of iterated integrals of quasimodular forms
We study the algebra of iterated integrals of quasimodular
forms for , which is the smallest extension of
the algebra of quasimodular forms, which is closed under
integration. We prove that is a polynomial algebra in
infinitely many variables, given by Lyndon words on certain monomials in
Eisenstein series. We also prove an analogous result for the
-subalgebra of of iterated
integrals of modular forms.Comment: v2, minor changes, to appear in Algebra and Number Theor
Families of eulerian functions involved in regularization of divergent polyzetas
Extending the Eulerian functions, we study their relationship with zeta
function of several variables. In particular, starting with Weierstrass
factorization theorem (and Newton-Girard identity) for the complex Gamma
function, we are interested in the ratios of and their
multiindexed generalization, we will obtain an analogue situation and draw some
consequences about a structure of the algebra of polyzetas values, by means of
some combinatorics of noncommutative rational series. The same combinatorial
frameworks also allow to study the independence of a family of eulerian
functions.Comment: preprin