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411.
J. Fleischer V. A. Smirnov O. V. Tarasov 《Zeitschrift fur Physik C Particles and Fields》1997,74(2):379-386
A method of calculating Feynman diagrams from their small momentum expansion [1] is extended to diagrams with zero mass thresholds. We start from the asymptotic expansion in large masses [2] (applied to the case when all $M_i^2$ are large compared to all momenta squared). Using dimensional regularization, a finite result is obtained in terms of powers of logarithms (describing the zero-threshold singularity) times power series in the momentum squared. Surprisingly, these latter ones represent functions, which not only have the expected physical “second threshold” but have a branchcut singularity as well below threshold at a mirror position. These can be understood as pseudothresholds corresponding to solutions of the Landau equations. In the spacelike region the imaginary parts from the various contributions cancel. For the two-loop examples with one mass M, in the timelike region for q2 ≈ M2 we obtain approximations of high precision. This will be of relevance in particular for the calculation of the decay Z → bb?in the m b = 0 approximation. 相似文献
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Avner Magen 《Discrete and Computational Geometry》2007,38(1):139-153
Let X be a subset of n points of the Euclidean space, and let 0 < ε < 1. A classical result of Johnson and Lindenstrauss [JL]
states that there is a projection of X onto a subspace of dimension
O(ε-2 log n) with distortion ≤ 1+ ε. We show a natural extension of the above result to a stronger preservation of the geometry
of finite spaces. By a k-fold increase
of the number of dimensions used compared with [JL], a good preservation of volumes and of distances between points and affine
spaces is achieved. Specifically, we show how to embed a subset of size n of the Euclidean space into a O(ε-2 log n)-dimensional Euclidean space, so that no set of size s ≤ k changes its volume by more than (1 + εs-1. Moreover, distances of points from affine hulls of sets of at most k - 1 points in the space do not change by more than
a factor of 1 + ε. A consequence of the above with k = 3 is that angles can be preserved using asymptotically the same number
of dimensions as the one used in [JL]. Our method can be applied to many problems with high-dimensional nature such as Projective
Clustering and Approximated Nearest Affine Neighbour Search. In particular, it shows a first polylogarithmic query time approximation
algorithm to the latter. We also show a structural application that for volume respecting embedding in the sense introduced
by Feige [F], the host space need not generally be of dimensionality greater than polylogarithmic in the size of the graph. 相似文献
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