At the most fundamental level, nature is described by the interaction of fundamental
particles. In order to further our understanding of the universe at this most fundamental
level, it is of vital importance that theory predictions are compared to measurements at
ever-increasing precision. The production of colour-singlet states, such as Higgs bosons
and electroweak bosons, is of particular interest. Not only do these particles play central
roles in electroweak symmetry breaking and the fate of the universe, but the theoretical
description of their production can be performed at unusually high orders in perturbation
theory, leading to high-precision predictions. The goal of this project is to perform cross
section calculations that allow us to make predictions for such processes under realistic
conditions measurable in experiments, at unprecedented next-to-next-to-next-to-leading
order precision. These leading-edge computations are, however, extremely demanding
and easily require several million CPU hours. Without the use of a HPC, the predictions
we aim to make simply cannot be made.
September 30, 2021–September 30, 2022
The type of calculation we want to perform cannot be done fully analytically. The
only way to describe realistic nal states with phase-space cuts similar to those used in
experiments is to integrate the phase space numerically. Due to the high dimensionality
of the phase spaces – up to 16 dimensions – quadrature methods are not an option.
Instead, we use the Monte Carlo numerical integration method. Even so, the cross section
contains many singularities with a highly non-trivial structure. These singularities are
handled according to the sector-improved residue subtraction scheme, as implemented
in the C++ code Stripper.
Unfortunately, due to theoretical issues described in the next subsection, we were thus
far unable to complete our calculation. These problems could not have been foreseen
and took a considerable amount of time to solve, which is the reason why this project
has been extended.
After solving the abovementioned issues, we were able to con rm results at lower pertur-
bative orders, all the way through the next-to-next-to-leading order. This includes re-
sults obtained by other groups and results obtained by ourselves using a well-established
alternative technique, which however is not applicable to the perturbative order we
ultimately wish to reach.
There were two main issues which have been solved during the time of the original
project: optimisation and technical biases.
At the start of the project, it soon became clear that to successfully perform the calcu-
lation using a reasonable amount of resources, it needed to be optimised signi cantly.
The biggest gain in speed has been due to an improvement to the implementation of
the sector-improved residue subtraction scheme. The subtraction scheme requires the
phase space to be decomposed so as to isolate all singularities before regulating them.
The previous implementation of this procedure sometimes created very strongly peaked
integrands, which are extremely di cult to integrate numerically. Once this problem
was identi ed, a suitable solution was devised and implemented, which sped up the code
to the point where our intended calculation was made feasible.
The second major issue was a systematic bias in the results caused by a necessary tech-
nical cut: due to the instability of
oating-point calculations involving small di erences
of large numbers, a so-called technical cut needs to be introduced to prevent the pro-
gram from sampling such problematic regions. If this cut is chosen too small, it does
not su ciently shield the calculation from numerical instabilities. If it is chosen too
large, it can introduce a bias. For the intended calculation, it turned out that the pre-
vious implementation of this cut could not be chosen to simultaneously stabilise the
calculation and keep it unbiased. It took signi cant testing and analysis of designated
runs to nd a suitable replacement cut that performs as required. The new version can
completely protect the calculation from the relevant numerical instabilities, without any
In the time between the previous status report, submitted as part of the project ex-
tension application, and the end of the original project, we were able to test the new
technical cut with a greater number of events, reducing the Monte Carlo integration
error and increasing our sensitivity to any biases still present. Within the improved
uncertainties, we were still unable to nd any leftover bias.
In that same period, we also substituted our implementation of certain mathematical
functions with high-precision approximations, which has sped up the evaluation of specifc parts of the calculation by a factor of up to 200.
Additional Project Information
DFG classification: 309
Software: The C++ library Stripper (private code developed by the PI and several collaborators)