34 research outputs found
Regularity of Edge Ideals and Their Powers
We survey recent studies on the Castelnuovo-Mumford regularity of edge ideals
of graphs and their powers. Our focus is on bounds and exact values of and the asymptotic linear function , for in terms of combinatorial data of the given graph Comment: 31 pages, 15 figure
Single Cut Integration
We present an analytic technique for evaluating single cuts for one-loop
integrands, where exactly one propagator is taken to be on shell. Our method
extends the double-cut integration formalism of one-loop amplitudes to the
single-cut case. We argue that single cuts give meaningful information about
amplitudes when taken at the integrand level. We discuss applications to the
computation of tadpole coefficients.Comment: v2: corrected typo in abstrac
On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes
We propose a first implementation of the integrand-reduction method for
two-loop scattering amplitudes. We show that the residues of the amplitudes on
multi-particle cuts are polynomials in the irreducible scalar products
involving the loop momenta, and that the reduction of the amplitudes in terms
of master integrals can be realized through polynomial fitting of the
integrand, without any apriori knowledge of the integral basis. We discuss how
the polynomial shapes of the residues determine the basis of master integrals
appearing in the final result. We present a four-dimensional constructive
algorithm that we apply to planar and non-planar contributions to the 4- and
5-point MHV amplitudes in N=4 SYM. The technique hereby discussed extends the
well-established analogous method holding for one-loop amplitudes, and can be
considered a preliminary study towards the systematic reduction at the
integrand-level of two-loop amplitudes in any gauge theory, suitable for their
automated semianalytic evaluation.Comment: 26 pages, 11 figure
Yang-Mills amplitude relations at loop level from non-adjacent BCFW shifts
This article studies methods to obtain relations for scattering amplitudes at
the loop level, with concrete examples at one loop. These methods originate in
the analysis of large so-called Britto-Cachazo-Feng-Witten shifts of tree level
amplitudes and loop level integrands. In particular BCFW shifts for particles
which are not color adjacent and some particular generalizations of this
situation are analyzed in some detail in four and higher dimensions. For
generic non-adjacent shifts our results are independent of loop order for
integrands and hold for generic minimally coupled gauge theories with possible
scalar potential and Yukawa terms. By a standard argument this result indicates
a generalization of the Bern-Carrasco-Johansson relations for tree level
amplitudes exists to the integrand at all loop levels. A concrete relation is
presented at one loop. Furthermore, inspired by results in QED it is shown that
the results on generalized BCFW shifts of tree level amplitudes imply relations
for the so-called rational, bubble and triangle terms of one loop amplitudes in
pure Yang-Mills theory. Bubble and triangle terms for instance are shown to
obey a five photon decoupling identity, while a three photon decoupling
identity is demonstrated for the rational terms. Along the same lines recently
conjectured relations for helicity equal amplitudes at one loop are shown to
generalize to helicity independent relations for the massive box coefficient of
the rational terms.Comment: 69 pages, 27 figure