12,980 research outputs found
Numerical evaluation of multi-loop integrals by sector decomposition
In a recent paper we have presented an automated subtraction method for
divergent multi-loop/leg integrals in dimensional regularisation which allows
for their numerical evaluation, and applied it to diagrams with massless
internal lines. Here we show how to extend this algorithm to Feynman diagrams
with massive propagators and arbitrary propagator powers. As applications, we
present numerical results for the master 2-loop 4-point topologies with massive
internal lines occurring in Bhabha scattering at two loops, and for the master
integrals of planar and non-planar massless double box graphs with two
off-shell legs. We also evaluate numerically some two-point functions up to 5
loops relevant for beta-function calculations, and a 3-loop 4-point function,
the massless on-shell planar triple box. Whereas the 4-point functions are
evaluated in non-physical kinematic regions, the results for the propagator
functions are valid for arbitrary kinematics.Comment: 15 pages latex, 11 eps figures include
Analytical expressions of 3 and 4-loop sunrise Feynman integrals and 4-dimensional lattice integrals
In this paper we continue the work begun in 2002 on the identification of the
analytical expressions of Feynman integrals which require the evaluation of
multiple elliptic integrals. We rewrite and simplify the analytical expression
of the 3-loop self-mass integral with three equal masses and on-shell external
momentum. We collect and analyze a number of results on double and triple
elliptic integrals. By using very high-precision numerical fits, for the first
time we are able to identify a very compact analytical expression for the
4-loop on-shell self-mass integral with 4 equal masses, that is one of the
master integrals of the 4-loop electron g-2. Moreover, we fit the analytical
expressions of some integrals which appear in lattice perturbation theory, and
in particular the 4-dimensional generalized Watson integral.Comment: 13 pages, 1 figure, LaTex; v2: some rephrasing of text; v3: reference
added, minor modifications; v4: checks of lattice integrals up to 2400
digits; some modifications of text; version accepted for publication in
IJMPA, needs the document class ws-ijmpa.cl
Numerical evaluation of loop integrals
We present a new method for the numerical evaluation of arbitrary loop
integrals in dimensional regularization. We first derive Mellin-Barnes integral
representations and apply an algorithmic technique, based on the Cauchy
theorem, to extract the divergent parts in the epsilon->0 limit. We then
perform an epsilon-expansion and evaluate the integral coefficients of the
expansion numerically. The method yields stable results in physical kinematic
regions avoiding intricate analytic continuations. It can also be applied to
evaluate both scalar and tensor integrals without employing reduction methods.
We demonstrate our method with specific examples of infrared divergent
integrals with many kinematic scales, such as two-loop and three-loop box
integrals and tensor integrals of rank six for the one-loop hexagon topology
Evaluating double and triple (?) boxes
A brief review of recent results on analytical evaluation of double-box
Feynman integrals is presented. First steps towards evaluation of massless
on-shell triple-box Feynman integrals within dimensional regularization are
described. The leading power asymptotic behaviour of the dimensionally
regularized massless on-shell master planar triple-box diagram in the Regge
limit is evaluated. The evaluation of the unexpanded master planar
triple box is outlined and explicit results for coefficients at 1/\ep^j,
j=2,...,6, are presented.Comment: 6 pages, LaTeX with axodraw.sty; Talk presented at the International
Symposium Radcor 2002 and Loops and Legs 2002 (September 8--13, Kloster Banz,
Germany); to appear in Nuclear Physics B (Proc. Suppl.); a wrong abstract is
replace
Evaluating multiloop Feynman integrals by Mellin-Barnes representation
The status of analytical evaluation of double and triple box diagrams is
characterized. The method of Mellin-Barnes representation as a tool to evaluate
master integrals in these problems is advocated. New MB representations for
massive on-shell double boxes with general powers of propagators are presented.Comment: 5 pages, Talk given at the 7th DESY workshop on Elementary Particle
Theory, "Loops and Legs in Quantum Field Theory", April 25-30, 2004,
Zinnowitz, Germany, to appear in the proceeding
A numerical method for NNLO calculations
A method to isolate the poles of dimensionally regulated multi-loop integrals
and to calculate the pole coefficients numerically is extended to be applicable
to phase space integrals as well.Comment: 5 pages, Talk presented at Radcor/Loops and Legs 2002, to appear in
the proceeding
Scattering AMplitudes from Unitarity-based Reduction Algorithm at the Integrand-level
SAMURAI is a tool for the automated numerical evaluation of one-loop
corrections to any scattering amplitudes within the dimensional-regularization
scheme. It is based on the decomposition of the integrand according to the
OPP-approach, extended to accommodate an implementation of the generalized
d-dimensional unitarity-cuts technique, and uses a polynomial interpolation
exploiting the Discrete Fourier Transform. SAMURAI can process integrands
written either as numerator of Feynman diagrams or as product of tree-level
amplitudes. We discuss some applications, among which the 6- and 8-photon
scattering in QED, and the 6-quark scattering in QCD. SAMURAI has been
implemented as a Fortran90 library, publicly available, and it could be a
useful module for the systematic evaluation of the virtual corrections oriented
towards automating next-to-leading order calculations relevant for the LHC
phenomenology.Comment: 35 pages, 7 figure
Calculation of Massless Feynman Integrals using Harmonic Sums
A method for the evaluation of the epsilon expansion of multi-loop massless
Feynman integrals is introduced. This method is based on the Gegenbauer
polynomial technique and the expansion of the Gamma function in terms of
harmonic sums. Algorithms for the evaluation of nested and harmonic sums are
used to reduce the expressions to get analytical or numerical results for the
expansion coefficients. Methods to increase the precision of numerical results
are discussed.Comment: 30 pages, 6 figures; Minor typos corrected, references added.
Published in Computer Physics Communication
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