193,567 research outputs found
The Two-Loop Master Integrals for
We compute the full set of two-loop Feynman integrals appearing in massless
two-loop four-point functions with two off-shell legs with the same invariant
mass. These integrals allow to determine the two-loop corrections to the
amplitudes for vector boson pair production at hadron colliders, , and thus to compute this process to next-to-next-to-leading order
accuracy in QCD. The master integrals are derived using the method of
differential equations, employing a canonical basis for the integrals. We
obtain analytical results for all integrals, expressed in terms of multiple
polylogarithms. We optimize our results for numerical evaluation by employing
functions which are real valued for physical scattering kinematics and allow
for an immediate power series expansion.Comment: 26 pages, results included as ancillary files. v2: minor typos
corrected, references added, published on JHE
Numerical Multi-Loop Calculations via Finite Integrals and One-Mass EW-QCD Drell-Yan Master Integrals
We study a recently-proposed approach to the numerical evaluation of
multi-loop Feynman integrals using available sector decomposition programs. As
our main example, we consider the two-loop integrals for the
corrections to Drell-Yan lepton production with up to one massive vector boson
in physical kinematics. As a reference, we evaluate these planar and non-planar
integrals by the method of differential equations through to weight five.
Choosing a basis of finite integrals for the numerical evaluation with SecDec3
leads to tremendous performance improvements and renders the otherwise
problematic seven-line topologies numerically accessible. As another example,
basis integrals for massless QCD three loop form factors are evaluated with
FIESTA4. Here, employing a basis of finite integrals results in an overall
speedup of more than an order of magnitude.Comment: 24 pages, 1 figure, 4 tables, 2 ancillary files with analytical
results; in v2: minor improvements in the text with additional references
added. v2 is the version published in JHE
Laws of large numbers and Langevin approximations for stochastic neural field equations
In this study we consider limit theorems for microscopic stochastic models of
neural fields. We show that the Wilson-Cowan equation can be obtained as the
limit in probability on compacts for a sequence of microscopic models when the
number of neuron populations distributed in space and the number of neurons per
population tend to infinity. Though the latter divergence is not necessary.
This result also allows to obtain limits for qualitatively different stochastic
convergence concepts, e.g., convergence in the mean. Further, we present a
central limit theorem for the martingale part of the microscopic models which,
suitably rescaled, converges to a centered Gaussian process with independent
increments. These two results provide the basis for presenting the neural field
Langevin equation, a stochastic differential equation taking values in a
Hilbert space, which is the infinite-dimensional analogue of the Chemical
Langevin Equation in the present setting. On a technical level we apply
recently developed law of large numbers and central limit theorems for
piecewise deterministic processes taking values in Hilbert spaces to a master
equation formulation of stochastic neuronal network models. These theorems are
valid for processes taking values in Hilbert spaces and by this are able to
incorporate spatial structures of the underlying model.Comment: 38 page
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