18 research outputs found
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
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
Elliptic multizetas and the elliptic double shuffle relations
We define an elliptic generating series whose coefficients, the elliptic
multizetas, are related to the elliptic analogues of multiple zeta values
introduced by Enriquez as the coefficients of his elliptic associator; both
sets of coefficients lie in , the ring of functions
on the Poincar\'e upper half-plane . The elliptic multizetas
generate a -algebra which is an elliptic analogue of
the algebra of multiple zeta values. Working modulo , we show that the
algebra decomposes into a geometric and an arithmetic part and
study the precise relationship between the elliptic generating series and the
elliptic associator defined by Enriquez. We show that the elliptic multizetas
satisfy a double shuffle type family of algebraic relations similar to the
double shuffle relations satisfied by multiple zeta values. We prove that these
elliptic double shuffle relations give all algebraic relations among elliptic
multizetas if (a) the classical double shuffle relations give all algebraic
relations among multiple zeta values and (b) the elliptic double shuffle Lie
algebra has a certain natural semi-direct product structure analogous to that
established by Enriquez for the elliptic Grothendieck-Teichm\"uller Lie
algebra.Comment: major revision, to appear in: Int. Math. Res. No
A Novel Algorithm for Nested Summation and Hypergeometric Expansions
We consider a class of sums over products of Z-sums whose arguments differ by
a symbolic integer. Such sums appear, for instance, in the expansion of Gauss
hypergeometric functions around integer indices that depend on a symbolic
parameter. We present a telescopic algorithm for efficiently converting these
sums into generalized polylogarithms, Z-sums, and cyclotomic harmonic sums for
generic values of this parameter. This algorithm is illustrated by computing
the double pentaladder integrals through ten loops, and a family of massive
self-energy diagrams through in dimensional regularization. We
also outline the general telescopic strategy of this algorithm, which we
anticipate can be applied to other classes of sums.Comment: 36 pages, 2 figures; v2: references added, typos corrected, improved
introduction and comparison with existing methods, matches published versio