8,727 research outputs found
Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
This is a sequel of our previous paper where we described an algorithm to
find a solution of differential equations for master integrals in the form of
an -expansion series with numerical coefficients. The algorithm is
based on using generalized power series expansions near singular points of the
differential system, solving difference equations for the corresponding
coefficients in these expansions and using matching to connect series
expansions at two neighboring points. Here we use our algorithm and the
corresponding code for our example of four-loop generalized sunset diagrams
with three massive and two massless propagators, in order to obtain new
analytical results. We analytically evaluate the master integrals at threshold,
, in an expansion in up to . With the help of
our code, we obtain numerical results for the threshold master integrals in an
-expansion with the accuracy of 6000 digits and then use the PSLQ
algorithm to arrive at analytical values. Our basis of constants is build from
bases of multiple polylogarithm values at sixth roots of unity.Comment: Discussion extende
The AdS(n) x S(n) x T(10-2n) BMN string at two loops
We calculate the two-loop correction to the dispersion relation for
worldsheet modes of the BMN string in AdS(n) x S(n) x T(10-2n) for n=2,3,5. For
the massive modes the result agrees with the exact dispersion relation derived
from symmetry considerations with no correction to the interpolating function
h. For the massless modes in AdS(3) x S(3) x T(4) however our result does not
match what one expects from the corresponding symmetry based analysis. We also
derive the S-matrix for massless modes up to the one-loop order. The scattering
phase is given by the massless limit of the Hernandez-Lopez phase. In addition
we compute a certain massless S-matrix element at two loops and show that it
vanishes suggesting that the two-loop phase in the massless sector is zero.Comment: 30 pages, 6 figures; v2: References and comment on type IIB added,
acknowledgements updated; v3: Comparison to proposed exact massless S-matrix
in sec 5.3 corrected. Only non-trivial phase appears at one loop. Additional
minor clarification
MATAD: a program package for the computation of MAssive TADpoles
In the recent years there has been an enormous development in the evaluation
of higher order quantum corrections. An essential ingredient in the practical
calculations is provided by vacuum diagrams, i.e. integrals without external
momenta. In this paper a program package is described which can deal with one-,
two- and three-loop vacuum integrals with one non-zero mass parameter. The
principle structure is introduced and the main parts of the package are
described in detail. Explicit examples demonstrate the fields of application.Comment: 37 pages, to be published in Comp. Phys. Commu
A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
We define the rigidity of a Feynman integral to be the smallest dimension
over which it is non-polylogarithmic. We argue that massless Feynman integrals
in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show
that this bound may be saturated for integrals that we call marginal: those
with (L+1)D/2 propagators in (even) D dimensions. We show that marginal Feynman
integrals in D dimensions generically involve Calabi-Yau geometries, and we
give examples of finite four-dimensional Feynman integrals in massless
theory that saturate our predicted bound in rigidity at all loop orders.Comment: 5+2 pages, 11 figures, infinite zoo of Calabi-Yau manifolds. v2
reflects minor changes made for publication. This version is authoritativ
Massive Nonplanar Two-Loop Maximal Unitarity
We explore maximal unitarity for nonplanar two-loop integrals with up to four
massive external legs. In this framework, the amplitude is reduced to a basis
of master integrals whose coefficients are extracted from maximal cuts. The
hepta-cut of the nonplanar double box defines a nodal algebraic curve
associated with a multiply pinched genus-3 Riemann surface. All possible
configurations of external masses are covered by two distinct topological
pictures in which the curve decomposes into either six or eight Riemann
spheres. The procedure relies on consistency equations based on vanishing of
integrals of total derivatives and Levi-Civita contractions. Our analysis
indicates that these constraints are governed by the global structure of the
maximal cut. Lastly, we present an algorithm for computing generalized cuts of
massive integrals with higher powers of propagators based on the Bezoutian
matrix method.Comment: 54 pages, 9 figures, v2: journal versio
Antenna subtraction with massive fermions at NNLO: Double real initial-final configurations
We derive the integrated forms of specific initial-final tree-level
four-parton antenna functions involving a massless initial-state parton and a
massive final-state fermion as hard radiators. These antennae are needed in the
subtraction terms required to evaluate the double real corrections to
hadronic production at the NNLO level stemming from the partonic
processes and .Comment: 24 pages, 1 figure, 1 Mathematica file attache
Reduction of one-massless-loop with scalar boxes in dimensions
All one-massless-loop Feynman diagrams could be written like a linear
combination of scalar boxes, triangles an bubbles in dimensions plus
rational terms. However, the four-point scalar integrals in dimensions
are free of infrared divergences. We are going to change the dimensions of the
scalar boxes and the using of this degree of freedom to simplify
the computation of coefficients of the decomposition.Comment: 17 page
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