947 research outputs found
A Jacobian module for disentanglements and applications to Mond's conjecture
Given a germ of holomorphic map from to ,
we define a module whose dimension over is an upper bound
for the -codimension of , with equality if is weighted
homogeneous. We also define a relative version of the module, for
unfoldings of . The main result is that if are nice
dimensions, then the dimension of over is an upper bound of
the image Milnor number of , with equality if and only if the relative
module is Cohen-Macaulay for some stable unfolding . In particular,
if is Cohen-Macaulay, then we have Mond's conjecture for .
Furthermore, if is quasi-homogeneous, then Mond's conjecture for is
equivalent to the fact that is Cohen-Macaulay. Finally, we observe
that to prove Mond's conjecture, it suffices to prove it in a suitable family
of examples.Comment: 19 page
Off-shell Currents and Color-Kinematics Duality
We elaborate on the color-kinematics duality for off-shell diagrams in gauge
theories coupled to matter, by investigating the scattering process , and show that the Jacobi relations for the kinematic numerators
of off-shell diagrams, built with Feynman rules in axial gauge, reduce to a
color-kinematics violating term due to the contributions of sub-graphs only.
Such anomaly vanishes when the four particles connected by the Jacobi relation
are on their mass shell with vanishing squared momenta, being either external
or cut particles, where the validity of the color-kinematics duality is
recovered. We discuss the role of the off-shell decomposition in the direct
construction of higher-multiplicity numerators satisfying color-kinematics
identity in four as well as in dimensions, for the latter employing the
Four Dimensional Formalism variant of the Four Dimensional Helicity scheme. We
provide explicit examples for the QCD process .Comment: Accepted version for publication in PLB. Manuscript extended: 19
pages, 15 figures; C/K duality for tree-level amplitudes in dimensional
regularization added; references added; title modifie
Adaptive Integrand Decomposition
We present a simplified variant of the integrand reduction algorithm for
multiloop scattering amplitudes in dimensions, which
exploits the decomposition of the integration momenta in parallel and
orthogonal subspaces, , where is the
dimension of the space spanned by the legs of the diagrams. We discuss the
advantages of a lighter polynomial division algorithm and how the orthogonality
relations for Gegenbauer polynomilas can be suitably used for carrying out the
integration of the irreducible monomials, which eliminates spurious integrals.
Applications to one- and two-loop integrals, for arbitrary kinematics, are
discussed.Comment: Conference Proceedings, Loops and Legs in Quantum Field Theory, 24-29
April 2016, Leipzig, German
Generalised Unitarity for Dimensionally Regulated Amplitudes
We present a novel set of Feynman rules and generalised unitarity
cut-conditions for computing one-loop amplitudes via d-dimensional integrand
reduction algorithm. Our algorithm is suited for analytic as well as numerical
result, because all ingredients turn out to have a four-dimensional
representation. We will apply this formalism to NLO QCD corrections.Comment: Presented at SILAFAE 2014, 24-28 Nov, Ruta N, Medellin, Colombi
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