460 research outputs found
Mathematical aspects of scattering amplitudes
In these lectures we discuss some of the mathematical structures that appear
when computing multi-loop Feynman integrals. We focus on a specific class of
special functions, the so-called multiple polylogarithms, and discuss introduce
their Hopf algebra structure. We show how these mathematical concepts are
useful in physics by illustrating on several examples how these algebraic
structures are useful to perform analytic computations of loop integrals, in
particular to derive functional equations among polylogarithms.Comment: 58 pages. Lectures presented at TASI 201
PolyLogTools - Polylogs for the masses
We review recent developments in the study of multiple polylogarithms,
including the Hopf algebra of the multiple polylogarithms and the symbol map,
as well as the construction of single valued multiple polylogarithms and
discuss an algorithm for finding fibration bases. We document how these
algorithms are implemented in the Mathematica package PolyLogTools and show how
it can be used to study the coproduct structure of polylogarithmic expressions
and how to compute iterated parametric integrals over polylogarithmic
expressions that show up in Feynman integal computations at low loop orders.Comment: Package URL: https://gitlab.com/pltteam/pl
Iteration of Planar Amplitudes in Maximally Supersymmetric Yang-Mills Theory at Three Loops and Beyond
We compute the leading-color (planar) three-loop four-point amplitude of N=4
supersymmetric Yang-Mills theory in 4 - 2 epsilon dimensions, as a Laurent
expansion about epsilon = 0 including the finite terms. The amplitude was
constructed previously via the unitarity method, in terms of two Feynman loop
integrals, one of which has been evaluated already. Here we use the
Mellin-Barnes integration technique to evaluate the Laurent expansion of the
second integral. Strikingly, the amplitude is expressible, through the finite
terms, in terms of the corresponding one- and two-loop amplitudes, which
provides strong evidence for a previous conjecture that higher-loop planar N =
4 amplitudes have an iterative structure. The infrared singularities of the
amplitude agree with the predictions of Sterman and Tejeda-Yeomans based on
resummation. Based on the four-point result and the exponentiation of infrared
singularities, we give an exponentiated ansatz for the maximally
helicity-violating n-point amplitudes to all loop orders. The 1/epsilon^2 pole
in the four-point amplitude determines the soft, or cusp, anomalous dimension
at three loops in N = 4 supersymmetric Yang-Mills theory. The result confirms a
prediction by Kotikov, Lipatov, Onishchenko and Velizhanin, which utilizes the
leading-twist anomalous dimensions in QCD computed by Moch, Vermaseren and
Vogt. Following similar logic, we are able to predict a term in the three-loop
quark and gluon form factors in QCD.Comment: 54 pages, 7 figures. v2: Added references, a few additional words
about large spin limit of anomalous dimensions. v3: Expanded Sect. IV.A on
multiloop ansatz; remark that form-factor prediction is now confirmed by
other work; minor typos correcte
The physics and the mixed Hodge structure of Feynman integrals
This expository text is an invitation to the relation between quantum field
theory Feynman integrals and periods. We first describe the relation between
the Feynman parametrization of loop amplitudes and world-line methods, by
explaining that the first Symanzik polynomial is the determinant of the period
matrix of the graph, and the second Symanzik polynomial is expressed in terms
of world-line Green's functions. We then review the relation between Feynman
graphs and variations of mixed Hodge structures. Finally, we provide an
algorithm for generating the Picard-Fuchs equation satisfied by the all equal
mass banana graphs in a two-dimensional space-time to all loop orders.Comment: v2: 34 pages, 5 figures. Minor changes. References added. String-math
2013 proceeding contributio
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