104 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
Iterated Binomial Sums and their Associated Iterated Integrals
We consider finite iterated generalized harmonic sums weighted by the
binomial in numerators and denominators. A large class of these
functions emerges in the calculation of massive Feynman diagrams with local
operator insertions starting at 3-loop order in the coupling constant and
extends the classes of the nested harmonic, generalized harmonic and cyclotomic
sums. The binomially weighted sums are associated by the Mellin transform to
iterated integrals over square-root valued alphabets. The values of the sums
for and the iterated integrals at lead to new
constants, extending the set of special numbers given by the multiple zeta
values, the cyclotomic zeta values and special constants which emerge in the
limit of generalized harmonic sums. We develop
algorithms to obtain the Mellin representations of these sums in a systematic
way. They are of importance for the derivation of the asymptotic expansion of
these sums and their analytic continuation to . The
associated convolution relations are derived for real parameters and can
therefore be used in a wider context, as e.g. for multi-scale processes. We
also derive algorithms to transform iterated integrals over root-valued
alphabets into binomial sums. Using generating functions we study a few aspects
of infinite (inverse) binomial sums.Comment: 62 pages Latex, 1 style fil
Multiple (inverse) binomial sums of arbitrary weight and depth and the all-order epsilon-expansion of generalized hypergeometric functions with one half-integer value of parameter
We continue the study of the construction of analytical coefficients of the
epsilon-expansion of hypergeometric functions and their connection with Feynman
diagrams. In this paper, we show the following results:
Theorem A: The multiple (inverse) binomial sums of arbitrary weight and depth
(see Eq. (1.1)) are expressible in terms of Remiddi-Vermaseren functions.
Theorem B: The epsilon expansion of a hypergeometric function with one
half-integer value of parameter (see Eq. (1.2)) is expressible in terms of the
harmonic polylogarithms of Remiddi and Vermaseren with coefficients that are
ratios of polynomials. Some extra materials are available via the www at this
http://theor.jinr.ru/~kalmykov/hypergeom/hyper.htmlComment: 24 pages, latex with amsmath and JHEP3.cls; v2: some typos corrected
and a few references added; v3: few references added
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
Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations
We outline a new algorithm to solve coupled systems of differential equations
in one continuous variable (resp. coupled difference equations in one
discrete variable ) depending on a small parameter : given such a
system and given sufficiently many initial values, we can determine the first
coefficients of the Laurent-series solutions in if they are
expressible in terms of indefinite nested sums and products. This systematic
approach is based on symbolic summation algorithms in the context of difference
rings/fields and uncoupling algorithms. The proposed method gives rise to new
interesting applications in connection with integration by parts (IBP) methods.
As an illustrative example, we will demonstrate how one can calculate the
-expansion of a ladder graph with 6 massive fermion lines
3-Loop Heavy Flavor Corrections in Deep-Inelastic Scattering with Two Heavy Quark Lines
We consider gluonic contributions to the heavy flavor Wilson coefficients at
3-loop order in QCD with two heavy quark lines in the asymptotic region . Here we report on the complete result in the case of two equal
masses for the massive operator matrix element ,
which contributes to the corresponding heavy flavor transition matrix element
in the variable flavor number scheme. Nested finite binomial sums and iterated
integrals over square-root valued alphabets emerge in the result for this
quantity in and -space, respectively. We also present results for the
case of two unequal masses for the flavor non-singlet OMEs and on the scalar
integrals ic case of , which were calculated without a further
approximation. The graphs can be expressed by finite nested binomial sums over
generalized harmonic sums, the alphabet of which contains rational letters in
the ratio .Comment: 10 pages LATEX, 1 Figure, Proceedings of Loops and Legs in Quantum
Field Theory, Weimar April 201
Large scale analytic calculations in quantum field theories
We present a survey on the mathematical structure of zero- and single scale
quantities and the associated calculation methods and function spaces in higher
order perturbative calculations in relativistic renormalizable quantum field
theories.Comment: 25 pages Latex, 1 style fil
3-loop Massive Contributions to the DIS Operator Matrix Element
Contributions to heavy flavour transition matrix elements in the variable
flavour number scheme are considered at 3-loop order. In particular a
calculation of the diagrams with two equal masses that contribute to the
massive operator matrix element is performed. In the Mellin
space result one finds finite nested binomial sums. In -space these sums
correspond to iterated integrals over an alphabet containing also square-root
valued letters.Comment: 4 pages, Contribution to the Proceedings of QCD '14, Montpellier,
July 201
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