S@M, a Mathematica Implementation of the SpinorFormalism Page: 3 of 52
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The singularity information can be extracted by defining amplitudes for suit-
able complex, yet on-shell, values of the external momenta - an idea that
stemmed from Witten's development of twistor string theory [63-66]. Generat-
ing complex momenta by modifying spinor variables, considered as fundamen-
tal objects, leads to new ways to exploit the kinematic properties of helicity
amplitudes. The new-born complex momenta have the property of preserving
overall momentum conservation and on-shell nature. Complex kinematics al-
low the exploration of singularities of on-shell amplitudes and the use of factor-
ization information to reconstruct tree amplitudes recursively from their poles.
The application of factorization in the on-shell method is realized through glu-
ing lower-point tree amplitudes to form higher points ones linked by on-shell
yet complex propagating particles. The construction of tree amplitudes via
on-shell recursion essentially amounts to a reversal of the collinear limit. This
is made possible by complexifying momenta in their spinorial representation,
and results in a quadratic recursion, the BCFW recurrence relation , which
works for massive particles [68-71] as well.
Complex kinematics are useful for the fulfillment of generalized unitarity con-
ditions as well. At one loop, generalized unitarity corresponds to requiring
more than two internal particles to be on-shell, such constraints cannot be
realized in general with real Minkowski momenta.
The application of unitarity as an on-shell method of calculation is based
on two principles: i) sewing tree amplitudes together to form one-loop am-
plitudes; ii) decomposing loop-amplitudes in terms of a basis of scalar loop-
integrals [72,73]. Matching the generalized cuts of the amplitude with the cuts
of basic integrals provides an efficient way to extract the rational coefficients
from the decomposition. The unitarity method [35, 36] provides a technique
for producing functions with the correct branch cuts in all channels , as
determined by products of tree amplitudes.
The use of four-dimensional states and momenta in the cuts enable the con-
struction of the (poly)logarithmic terms in the amplitudes, but generically
drops rational terms, which have to be recovered independently.
More recent improvements to the unitarity method  use complex momenta
within generalized unitarity, allowing for a simple and purely algebraic deter-
mination of box integral coefficients from quadruple-cuts. Using double- and
triple-unitarity cuts have led to very efficient techniques for extracting trian-
gle and bubble integral coefficients analytically [42, 48, 50, 75]. In particular
in [42, 48, 50] the phase-space integration has been reduced to the extraction
of residues in spinor variables and, at the occurrence, to trivial Feynman-
parametric integration. This approach has been used to compute analytically
some contributions to the six-gluon amplitude [42, 48], and the calculation
of the complete six-photon amplitudes , whose cut-constructibility was
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Maitre, D. & Mastrolia, P. S@M, a Mathematica Implementation of the SpinorFormalism, article, November 2, 2007; [Menlo Park, California]. (https://digital.library.unt.edu/ark:/67531/metadc890820/m1/3/: accessed April 20, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.