7,908 research outputs found
On the status of expansion by regions
We discuss the status of expansion by regions, i.e. a well-known strategy to
obtain an expansion of a given multiloop Feynman integral in a given limit
where some kinematic invariants and/or masses have certain scaling measured in
powers of a given small parameter. Using the Lee-Pomeransky parametric
representation, we formulate the corresponding prescriptions in a simple
geometrical language and make a conjecture that they hold even in a much more
general case. We prove this conjecture in some partial cases and illustrate
them in a simple example.Comment: Published version: presentation improved, Section 7 delete
Evaluating single-scale and/or non-planar diagrams by differential equations
We apply a recently suggested new strategy to solve differential equations
for Feynman integrals. We develop this method further by analyzing asymptotic
expansions of the integrals. We argue that this allows the systematic
application of the differential equations to single-scale Feynman integrals.
Moreover, the information about singular limits significantly simplifies
finding boundary constants for the differential equations. To illustrate these
points we consider two families of three-loop integrals. The first are
form-factor integrals with two external legs on the light cone. We introduce
one more scale by taking one more leg off-shell, . We analytically
solve the differential equations for the master integrals in a Laurent
expansion in dimensional regularization with . Then we show
how to obtain analytic results for the corresponding one-scale integrals in an
algebraic way. An essential ingredient of our method is to match solutions of
the differential equations in the limit of small to our results at
and to identify various terms in these solutions according to
expansion by regions. The second family consists of four-point non-planar
integrals with all four legs on the light cone. We evaluate, by differential
equations, all the master integrals for the so-called graph consisting of
four external vertices which are connected with each other by six lines. We
show how the boundary constants can be fixed with the help of the knowledge of
the singular limits. We present results in terms of harmonic polylogarithms for
the corresponding seven master integrals with six propagators in a Laurent
expansion in up to weight six.Comment: 27 pages, 2 figure
Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
This is a sequel of our previous paper where we described an algorithm to
find a solution of differential equations for master integrals in the form of
an -expansion series with numerical coefficients. The algorithm is
based on using generalized power series expansions near singular points of the
differential system, solving difference equations for the corresponding
coefficients in these expansions and using matching to connect series
expansions at two neighboring points. Here we use our algorithm and the
corresponding code for our example of four-loop generalized sunset diagrams
with three massive and two massless propagators, in order to obtain new
analytical results. We analytically evaluate the master integrals at threshold,
, in an expansion in up to . With the help of
our code, we obtain numerical results for the threshold master integrals in an
-expansion with the accuracy of 6000 digits and then use the PSLQ
algorithm to arrive at analytical values. Our basis of constants is build from
bases of multiple polylogarithm values at sixth roots of unity.Comment: Discussion extende
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