586 research outputs found
Feynman integrals and iterated integrals of modular forms
In this paper we show that certain Feynman integrals can be expressed as
linear combinations of iterated integrals of modular forms to all orders in the
dimensional regularisation parameter . We discuss explicitly the
equal mass sunrise integral and the kite integral. For both cases we give the
alphabet of letters occurring in the iterated integrals. For the sunrise
integral we present a compact formula, expressing this integral to all orders
in as iterated integrals of modular forms.Comment: 53 pages, v2: section on relation to Gamma_1(6) added, v3: section on
the choice of the periods adde
The two-loop sunrise graph with arbitrary masses
We discuss the analytical solution of the two-loop sunrise graph with
arbitrary non-zero masses in two space-time dimensions. The analytical result
is obtained by solving a second-order differential equation. The solution
involves elliptic integrals and in particular the solutions of the
corresponding homogeneous differential equation are given by periods of an
elliptic curve.Comment: 24 page
From elliptic curves to Feynman integrals
In this talk we discuss Feynman integrals which are related to elliptic
curves. We show with the help of an explicit example that in the set of master
integrals more than one elliptic curve may occur. The technique of maximal cuts
is a useful tool to identify the elliptic curves. By a suitable transformation
of the master integrals the system of differential equations for our example
can be brought into a form linear in , where the
-term is strictly lower-triangular. This system is easily solved
in terms of iterated integrals.Comment: 11 pages, talk given at Loops and Legs 201
The sunrise integral and elliptic polylogarithms
We summarize recent computations with a class of elliptic generalizations of
polylogarithms, arising from the massive sunrise integral. For the case of
arbitrary masses we obtain results in two and four space-time dimensions. The
iterated integral structure of our functions allows us to furthermore compute
the equal mass case to arbitrary order.Comment: talk given at Loops and Legs 201
The iterated structure of the all-order result for the two-loop sunrise integral
We present a method to compute the Laurent expansion of the two-loop sunrise
integral with equal non-zero masses to arbitrary order in the dimensional
regularisation . This is done by introducing a class of functions
(generalisations of multiple polylogarithms to include the elliptic case) and
by showing that all integrations can be carried out within this class of
functions.Comment: 20 pages, v2: version to be publishe
The Elliptic Sunrise
In this talk, we discuss our recent computation of the two-loop sunrise
integral with arbitrary non-zero particle masses. In two space-time dimensions,
we arrive at a result in terms of elliptic dilogarithms. Near four space-time
dimensions, we obtain a result which furthermore involves elliptic
generalizations of Clausen and Glaisher functions.Comment: talk at the Workshop on Multiple Zeta Values, Modular Forms and
Elliptic Motives II, December 2014, ICMAT Madri
The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case
We present the result for the finite part of the two-loop sunrise integral
with unequal masses in four space-time dimensions in terms of the -part and the -part of the
sunrise integral around two space-time dimensions. The latter two integrals are
given in terms of elliptic generalisations of Clausen and Glaisher functions.
Interesting aspects of the result for the -part of
the sunrise integral around two space-time dimensions are the occurrence of
depth two elliptic objects and the weights of the individual terms.Comment: 31 page
The sunrise integral around two and four space-time dimensions in terms of elliptic polylogarithms
In this talk we discuss the solution for the sunrise integral around two and
four space-time dimensions in terms of a generalised elliptic version of the
multiple polylogarithms. In two space-time dimensions we obtain a sum of three
elliptic dilogarithms. The arguments of the elliptic dilogarithms have a nice
geometric interpretation. In four space-time dimensions the sunrise integral
can be expressed with the - and -solution around two
dimensions, mass derivatives thereof and simpler terms.Comment: 5 pages, no figures, proceeding to the conference 'Matter To The
Deepest', Sept 14-18, 2015, Ustro\'n, Polan
The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
We compute systematically for the planar double box Feynman integral relevant
to top pair production with a closed top loop the Laurent expansion in the
dimensional regularisation parameter . This is done by
transforming the system of differential equations for this integral and all its
sub-topologies to a form linear in , where the
-part is strictly lower triangular. This system is easily solved
order by order in the dimensional regularisation parameter . This
is an example of an elliptic multi-scale integral involving several elliptic
sub-topologies. Our methods are applicable to similar problems.Comment: 6 pages, version to be publishe
A walk on sunset boulevard
A walk on sunset boulevard can teach us about transcendental functions
associated to Feynman diagrams. On this guided tour we will see multiple
polylogarithms, differential equations and elliptic curves. A highlight of the
tour will be the generalisation of the polylogarithms to the elliptic setting
and the all-order solution for the sunset integral in the equal mass case.Comment: 10 pages, talk given at Radcor 201
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