One of the main challenges in simulations on Lefschetz thimbles is the
computation of the relative weights of contributing thimbles. In this paper we
propose a solution to that problem by means of computing those weights using a
reweighting procedure. Besides we present recipes for finding parametrizations
of thimbles and anti-thimbles for a given theory. Moreover, we study some
approaches to combine the Lefschetz thimble method with the Complex Langevin
evolution. Our numerical investigations are carried out by using toy models
among which we consider a one-site z^4 model as well as a U(1) one-link model.
Precision phenomenology at the LHC requires accounting for both higher-order
QCD and electroweak corrections as well as for photon-initiated subprocesses.
Building upon the recent NNPDF3.1 fit, in this work the photon content of the
proton is determined within a global analysis supplemented by the LUXqed
constraint relating the photon PDF to lepton-proton scattering structure
functions: NNPDF3.1luxQED. The uncertainties on the resulting photon PDF are at
the level of a few percent, with photons carrying up to 0.5% of the proton's
momentum. We study the phenomenological implications of NNPDF3.1luxQED at the
LHC for Drell-Yan, vector boson pair, top quark pair, and Higgs plus vector
boson production. We find that photon-initiated contributions can be
significant for many processes, leading to corrections of up to 20%. Our
results represent a state-of-the-art determination of the partonic structure of
the proton including its photon component.
Establishing the completeness of a Bethe Ansatz solution for an exactly
solved model is a perennial challenge, which is typically approached on a case
by case basis. For the rational, spin-1/2 Richardson--Gaudin system it will be
argued that, for generic values of the system's coupling parameters, the Bethe
states are complete. This method does not depend on knowledge of the
distribution of Bethe roots, such as a string hypothesis, and is generalisable
to a wider class of systems.