44 research outputs found
On hyperlogarithms and Feynman integrals with divergences and many scales
It was observed that hyperlogarithms provide a tool to carry out Feynman
integrals. So far, this method has been applied successfully to finite
single-scale processes. However, it can be employed in more general situations.
We give examples of integrations of three- and four-point integrals in
Schwinger parameters with non-trivial kinematic dependence, involving setups
with off-shell external momenta and differently massive internal propagators.
The full set of Feynman graphs admissible to parametric integration is not yet
understood and we discuss some counterexamples to the crucial property of
linear reducibility. Furthermore we clarify how divergent integrals can be
approached in dimensional regularization with this algorithm.Comment: 26 pages, 11 figures, 2 tables, explicit results in ancillary file
"results" and on http://www.math.hu-berlin.de/~panzer/ (version as in JHEP;
link corrected
Feynman integrals via hyperlogarithms
This talk summarizes recent developments in the evaluation of Feynman
integrals using hyperlogarithms. We discuss extensions of the original method,
new results that were obtained with this approach and point out current
problems and future directions.Comment: 8 pages, 5 figures, Proceedings of "Loops & Legs 2014", Weimar
(Germany), April 27 -- May
Graphical functions in parametric space
Graphical functions are positive functions on the punctured complex plane
which arise in quantum field theory. We generalize
a parametric integral representation for graphical functions due to Lam, Lebrun
and Nakanishi, which implies the real analyticity of graphical functions.
Moreover we prove a formula that relates graphical functions of planar dual
graphs.Comment: v2: extended introduction, minor changes in notation and correction
of misprint
Feynman integral relations from parametric annihilators
We study shift relations between Feynman integrals via the Mellin transform
through parametric annihilation operators. These contain the momentum space IBP
relations, which are well-known in the physics literature. Applying a result of
Loeser and Sabbah, we conclude that the number of master integrals is computed
by the Euler characteristic of the Lee-Pomeransky polynomial. We illustrate
techniques to compute this Euler characteristic in various examples and compare
it with numbers of master integrals obtained in previous works.Comment: v2: new section 3.1 added, several misprints corrected and additional
remark
Hepp's bound for Feynman graphs and matroids
We study a rational matroid invariant, obtained as the tropicalization of the
Feynman period integral. It equals the volume of the polar of the matroid
polytope and we give efficient formulas for its computation. This invariant is
proven to respect all known identities of Feynman integrals for graphs. We
observe a strong correlation between the tropical and transcendental integrals,
which yields a method to approximate unknown Feynman periods.Comment: 26 figures, comments very welcom
Manifestly Dual-Conformal Loop Integration
Local, manifestly dual-conformally invariant loop integrands are now known
for all finite quantities associated with observables in planar, maximally
supersymmetric Yang-Mills theory through three loops. These representations,
however, are not infrared-finite term by term and therefore require
regularization; and even using a regulator consistent with dual-conformal
invariance, ordinary methods of loop integration would naively obscure this
symmetry. In this work, we show how any planar loop integral through at least
two loops can be systematically regulated and evaluated directly in terms of
strictly finite, manifestly dual-conformal Feynman-parameter integrals. We
apply these methods to the case of the two-loop ratio and remainder functions
for six particles, reproducing the known results in terms of individually
regulated local loop integrals, and we comment on some of the novelties that
arise for this regularization scheme not previously seen at one loop.Comment: 64 pages; 4 figures; complete details of the concrete examples are
provided in ancillary files. Typos fixed and similar improvements in v