2,006 research outputs found
On the Numerical Evaluation of Loop Integrals With Mellin-Barnes Representations
An improved method is presented for the numerical evaluation of multi-loop
integrals in dimensional regularization. The technique is based on
Mellin-Barnes representations, which have been used earlier to develop
algorithms for the extraction of ultraviolet and infrared divergencies. The
coefficients of these singularities and the non-singular part can be integrated
numerically. However, the numerical integration often does not converge for
diagrams with massive propagators and physical branch cuts. In this work,
several steps are proposed which substantially improve the behavior of the
numerical integrals. The efficacy of the method is demonstrated by calculating
several two-loop examples, some of which have not been known before.Comment: 13 pp. LaTe
Signatures of unstable semiclassical trajectories in tunneling
It was found recently that processes of multidimensional tunneling are
generally described at high energies by unstable semiclassical trajectories. We
study two observational signatures related to the instability of trajectories.
First, we find an additional power-law dependence of the tunneling probability
on the semiclassical parameter as compared to the standard case of potential
tunneling. The second signature is substantial widening of the probability
distribution over final-state quantum numbers. These effects are studied using
modified semiclassical technique which incorporates stabilization of the
tunneling trajectories. The technique is derived from first principles. We
obtain expressions for the inclusive and exclusive tunneling probabilities in
the case of unstable semiclassical trajectories. We also investigate the "phase
transition" between the cases of stable and unstable trajectories across
certain "critical" value of energy. Finally, we derive the relation between the
semiclassical probabilities of tunneling from the low-lying and highly excited
initial states. This puts on firm ground a conjecture made previously in the
semiclassical description of collision-induced tunneling in field theory.Comment: Journal version; 48 pages, 16 figure
On the Vacuum energy of a Color Magnetic Vortex
We calculate the one loop gluon vacuum energy in the background of a color
magnetic vortex for SU(2) and SU(3). We use zeta functional regularization to
obtain analytic expressions suitable for numerical treatment. The momentum
integration is turned to the imaginary axis and fast converging sums/integrals
are obtained. We investigate numerically a number of profiles of the
background. In each case the vacuum energy turns out to be positive increasing
in this way the complete energy and making the vortex configuration less
stable. In this problem bound states (tachyonic modes) are present for all
investigated profiles making them intrinsically unstable.Comment: 28 pages, 4 figure
Reduction schemes for one-loop tensor integrals
We present new methods for the evaluation of one-loop tensor integrals which
have been used in the calculation of the complete electroweak one-loop
corrections to e+ e- -> 4 fermions. The described methods for 3-point and
4-point integrals are, in particular, applicable in the case where the
conventional Passarino-Veltman reduction breaks down owing to the appearance of
Gram determinants in the denominator. One method consists of different variants
for expanding tensor coefficients about limits of vanishing Gram determinants
or other kinematical determinants, thereby reducing all tensor coefficients to
the usual scalar integrals. In a second method a specific tensor coefficient
with a logarithmic integrand is evaluated numerically, and the remaining
coefficients as well as the standard scalar integral are algebraically derived
from this coefficient. For 5-point tensor integrals, we give explicit formulas
that reduce the corresponding tensor coefficients to coefficients of 4-point
integrals with tensor rank reduced by one. Similar formulas are provided for
6-point functions, and the generalization to functions with more internal
propagators is straightforward. All the presented methods are also applicable
if infrared (soft or collinear) divergences are treated in dimensional
regularization or if mass parameters (for unstable particles) become complex.Comment: 55 pages, latex, some references updated and few comments added,
version to appear in Nucl. Phys.
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