13,315 research outputs found
Computer algebra tools for Feynman integrals and related multi-sums
In perturbative calculations, e.g., in the setting of Quantum Chromodynamics
(QCD) one aims at the evaluation of Feynman integrals. Here one is often faced
with the problem to simplify multiple nested integrals or sums to expressions
in terms of indefinite nested integrals or sums. Furthermore, one seeks for
solutions of coupled systems of linear differential equations, that can be
represented in terms of indefinite nested sums (or integrals). In this article
we elaborate the main tools and the corresponding packages, that we have
developed and intensively used within the last 10 years in the course of our
QCD-calculations
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
Towards a four-loop form factor
The four-loop, two-point form factor contains the first non-planar correction
to the lightlike cusp anomalous dimension. This anomalous dimension is a
universal function which appears in many applications. Its planar part in N = 4
SYM is known, in principle, exactly from AdS/CFT and integrability while its
non-planar part has been conjectured to vanish. The integrand of the form
factor of the stress-tensor multiplet in N = 4 SYM including the non-planar
part was obtained in previous work. We parametrise the difficulty of
integrating this integrand. We have obtained a basis of master integrals for
all integrals in the four-loop, two-point class in two ways. First, we computed
an IBP reduction of the integrand of the N = 4 form factor using massive
computer algebra (Reduze). Second, we computed a list of master integrals based
on methods of the Mint package, suitably extended using Macaulay2 / Singular.
The master integrals obtained in both ways are consistent with some minor
exceptions. The second method indicates that the master integrals apply beyond
N = 4 SYM, in particular to QCD. The numerical integration of several of the
master integrals will be reported and remaining obstacles will be outlinedComment: 9 Pages, Radcor/Loopfest 2015 Proceeding
A novel approach to integration by parts reduction
Integration by parts reduction is a standard component of most modern
multi-loop calculations in quantum field theory. We present a novel strategy
constructed to overcome the limitations of currently available reduction
programs based on Laporta's algorithm. The key idea is to construct algebraic
identities from numerical samples obtained from reductions over finite fields.
We expect the method to be highly amenable to parallelization, show a low
memory footprint during the reduction step, and allow for significantly better
run-times.Comment: 4 pages. Version 2 is the final, published version of this articl
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
Lie point symmetries and ODEs passing the Painlev\'e test
The Lie point symmetries of ordinary differential equations (ODEs) that are
candidates for having the Painlev\'e property are explored for ODEs of order . Among the 6 ODEs identifying the Painlev\'e transcendents only
, and have nontrivial symmetry algebras and that only
for very special values of the parameters. In those cases the transcendents can
be expressed in terms of simpler functions, i.e. elementary functions,
solutions of linear equations, elliptic functions or Painlev\'e transcendents
occurring at lower order. For higher order or higher degree ODEs that pass the
Painlev\'e test only very partial classifications have been published. We
consider many examples that exist in the literature and show how their symmetry
groups help to identify those that may define genuinely new transcendents
A toolbox to solve coupled systems of differential and difference equations
We present algorithms to solve coupled systems of linear differential
equations, arising in the calculation of massive Feynman diagrams with local
operator insertions at 3-loop order, which do {\it not} request special choices
of bases. Here we assume that the desired solution has a power series
representation and we seek for the coefficients in closed form. In particular,
if the coefficients depend on a small parameter \ep (the dimensional
parameter), we assume that the coefficients themselves can be expanded in
formal Laurent series w.r.t.\ \ep and we try to compute the first terms in
closed form. More precisely, we have a decision algorithm which solves the
following problem: if the terms can be represented by an indefinite nested
hypergeometric sum expression (covering as special cases the harmonic sums,
cyclotomic sums, generalized harmonic sums or nested binomial sums), then we
can calculate them. If the algorithm fails, we obtain a proof that the terms
cannot be represented by the class of indefinite nested hypergeometric sum
expressions. Internally, this problem is reduced by holonomic closure
properties to solving a coupled system of linear difference equations. The
underlying method in this setting relies on decoupling algorithms, difference
ring algorithms and recurrence solving. We demonstrate by a concrete example
how this algorithm can be applied with the new Mathematica package
\texttt{SolveCoupledSystem} which is based on the packages \texttt{Sigma},
\texttt{HarmonicSums} and \texttt{OreSys}. In all applications the
representation in -space is obtained as an iterated integral representation
over general alphabets, generalizing Poincar\'{e} iterated integrals
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