201 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

### Higgs production in bottom-quark fusion to third order in the strong coupling

We present the inclusive cross section at next-to-next-to-next-to-leading
order (N$^3$LO) in perturbative QCD for the production of a Higgs boson via
bottom-quark fusion. We employ the five-flavour scheme, treating the bottom
quark as a massless parton while retaining a non-vanishing Yukawa coupling to
the Higgs boson. We find that the dependence of the hadronic cross section on
the renormalisation and factorisation scales is substantially reduced. For
judicious choices of the scales the perturbative expansion of the cross section
shows a convergent behaviour. We present results for the N$^3$LO cross section
at various collider energies. In comparison to the cross section obtained from
the Santander-matching of the four and five-flavour schemes we predict a
slightly higher cross section, though the two predictions are consistent within
theoretical uncertainties.Comment: 3 pretty plots with pretty colours. Published versio

### Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral

We introduce a class of iterated integrals that generalize multiple
polylogarithms to elliptic curves. These elliptic multiple polylogarithms are
closely related to similar functions defined in pure math- ematics and string
theory. We then focus on the equal-mass and non-equal-mass sunrise integrals,
and we develop a formalism that enables us to compute these Feynman integrals
in terms of our iterated integrals on elliptic curves. The key idea is to use
integration-by-parts identities to identify a set of integral kernels, whose
precise form is determined by the branch points of the integral in question.
These kernels allow us to express all iterated integrals on an elliptic curve
in terms of them. The flexibility of our approach leads us to expect that it
will be applicable to a large variety of integrals in high-energy physics.Comment: 22 page

### Rational terms of UV origin to all loop orders

Numerical approaches to computations typically reconstruct the numerators of
Feynman diagrams in four dimensions. In doing so, certain rational terms
arising from the (D-4)-dimensional part of the numerator multiplying
ultraviolet (UV) poles in dimensional regularisation are not captured and need
to be obtained by other means. At one-loop these rational terms of UV origin
can be computed from a set of process-independent Feynman rules. Recently, it
was shown that this approach can be extended to two loops. In this paper, we
show that to all loop orders it is possible to compute rational terms of UV
origin through process-independent vertices that are polynomial in masses and
momenta.Comment: 19 page

### Feynman integrals in two dimensions and single-valued hypergeometric functions

We show that all Feynman integrals in two Euclidean dimensions with massless
propagators and arbitrary non-integer propagator powers can be expressed in
terms of single-valued analogues of Aomoto-Gelfand hypergeometric functions.
The latter can themselves be written as bilinears of hypergeometric functions,
with coefficients that are intersection numbers in a twisted homology group. As
an application, we show that all one-loop integrals in two dimensions with
massless propagators can be written in terms of Lauricella $F_D^{(r)}$
functions, while the $L$-loop ladder integrals are related to the generalised
hypergeometric ${}_{L+1}F_L$ functions.Comment: 44 pages, 11 figure

### A double integral of dlog forms which is not polylogarithmic

Feynman integrals are central to all calculations in perturbative Quantum
Field Theory. They often give rise to iterated integrals of dlog-forms with
algebraic arguments, which in many cases can be evaluated in terms of multiple
polylogarithms. This has led to certain folklore beliefs in the community
stating that all such integrals evaluate to polylogarithms. Here we discuss a
concrete example of a double iterated integral of two dlog-forms that evaluates
to a period of a cusp form. The motivic versions of these integrals are shown
to be algebraically independent from all multiple polylogarithms evaluated at
algebraic arguments. From a mathematical perspective, we study a mixed elliptic
Hodge structure arising from a simple geometric configuration in
$\mathbb{P}^2$, consisting of a modular plane elliptic curve and a set of lines
which meet it at torsion points, which may provide an interesting worked
example from the point of view of periods, extensions of motives, and
L-functions.Comment: 25 pages, 4 figures. To appear in the proceedings of
"Mathemamplitudes", held in Padova in December 201

### Multi-Regge kinematics and the scattering equations

We study the solutions to the scattering equations in various
quasi-multi-Regge regimes where the produced particles are ordered in rapidity.
We observe that in all cases the solutions to the scattering equations admit
the same hierarchy as the rapidity ordering, and we conjecture that this
behaviour holds independently of the number of external particles. In
multi-Regge limit, where the produced particles are strongly ordered in
rapidity, we determine exactly all solutions to the scattering equations that
contribute to the Cachazo-He-Yuan (CHY) formula for gluon scattering in this
limit. When the CHY formula is localised on these solutions, it reproduces the
expected factorisation of tree-level amplitudes in terms of impact factors and
Lipatov vertices. We also investigate amplitudes in various quasi-MRK. While in
these cases we cannot determine the solutions to the scattering equations
exactly, we show that again our conjecture combined with the CHY formula
implies the factorisation of the amplitude into universal buildings blocks for
which we obtain a CHY-type representation.Comment: 40 pages, 1 figur

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