3,655 research outputs found
Nested Integrals and Rationalizing Transformations
A brief overview of some computer algebra methods for computations with
nested integrals is given. The focus is on nested integrals over integrands
involving square roots. Rewrite rules for conversion to and from associated
nested sums are discussed. We also include a short discussion comparing the
holonomic systems approach and the differential field approach. For
simplification to rational integrands, we give a comprehensive list of
univariate rationalizing transformations, including transformations tuned to
map the interval bijectively to itself.Comment: manuscript of 25 February 2021, in "Anti-Differentiation and the
Calculation of Feynman Amplitudes", Springe
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
Light-Cone Expansion of the Dirac Sea in the Presence of Chiral and Scalar Potentials
We study the Dirac sea in the presence of external chiral and
scalar/pseudoscalar potentials. In preparation, a method is developed for
calculating the advanced and retarded Green's functions in an expansion around
the light cone. For this, we first expand all Feynman diagrams and then
explicitly sum up the perturbation series. The light-cone expansion expresses
the Green's functions as an infinite sum of line integrals over the external
potential and its partial derivatives.
The Dirac sea is decomposed into a causal and a non-causal contribution. The
causal contribution has a light-cone expansion which is closely related to the
light-cone expansion of the Green's functions; it describes the singular
behavior of the Dirac sea in terms of nested line integrals along the light
cone. The non-causal contribution, on the other hand, is, to every order in
perturbation theory, a smooth function in position space.Comment: 59 pages, LaTeX (published version
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
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
About calculation of massless and massive Feynman integrals
We report some results of calculations of massless and massive Feynman
integrals particularly focusing on difference equations for coefficients of for
their series expansionsComment: 51 pages; contribution to the proceedings of Helmholtz International
Summer School "Quantum Field Theory at the Limits: from Strong Fields to
Heavy Quarks" (July 22 - August 2, 2019; Dubna, Russia
Generating function for web diagrams
We present the description of the exponentiated diagrams in terms of
generating function within the universal diagrammatic technique. In particular,
we show the exponentiation of the gauge theory amplitudes involving products of
an arbitrary number of Wilson lines of arbitrary shapes, which generalizes the
concept of web diagrams. The presented method gives a new viewpoint on the web
diagrams and proves the non-Abelian exponentiation theorem.Comment: 8 pages, 3 figures; version accepted by PR
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