139 research outputs found
The hexagon Wilson loop and the BDS ansatz for the six-gluon amplitude
As a test of the gluon scattering amplitude/Wilson loop duality, we evaluate
the hexagonal light-like Wilson loop at two loops in N=4 super Yang-Mills
theory. We compare its finite part to the Bern-Dixon-Smirnov (BDS) conjecture
for the finite part of the six-gluon amplitude. We find that the two
expressions have the same behavior in the collinear limit, but they differ by a
non-trivial function of the three (dual) conformally invariant variables. This
implies that either the BDS conjecture or the gluon amplitude/Wilson loop
duality fails for the six-gluon amplitude, starting from two loops. Our results
are in qualitative agreement with the analysis of Alday and Maldacena of
scattering amplitudes with infinitely many external gluons.Comment: 8 pages, 3 figures; typos correcte
Generalized unitarity for N=4 super-amplitudes
We develop a manifestly supersymmetric version of the generalized unitarity
cut method for calculating scattering amplitudes in N=4 SYM theory. We
illustrate the power of this method by computing the one-loop n-point NMHV
super-amplitudes. The result confirms two conjectures which we made in
arXiv:0807.1095 [hep-th]. Firstly, we derive the compact, manifestly dual
superconformally covariant form of the NMHV tree amplitudes for arbitrary
number and types of external particles. Secondly, we show that the ratio of the
one-loop NMHV to the MHV amplitude is dual conformal invariant.Comment: 41 pages, 9 figure
Dual superconformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory
We argue that the scattering amplitudes in the maximally supersymmetric N=4
super-Yang-Mills theory possess a new symmetry which extends the previously
discovered dual conformal symmetry. To reveal this property we formulate the
scattering amplitudes as functions in the appropriate dual superspace.
Rewritten in this form, all tree-level MHV and next-to-MHV amplitudes exhibit
manifest dual superconformal symmetry. We propose a new, compact and Lorentz
covariant formula for the tree-level NMHV amplitudes for arbitrary numbers and
types of external particles. The dual conformal symmetry is broken at loop
level by infrared divergences. However, we provide evidence that the anomalous
contribution to the MHV and NMHV superamplitudes is the same and, therefore,
their ratio is a dual conformal invariant function. We identify this function
by an explicit calculation of the six-particle amplitudes at one loop. We
conjecture that these properties hold for all, MHV and non-MHV, superamplitudes
in N=4 SYM both at weak and at strong coupling.Comment: 58 page
The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N=4 SYM
We provide an analytic formula for the (rescaled) one-loop scalar hexagon
integral with all external legs massless, in terms of classical
polylogarithms. We show that this integral is closely connected to two
integrals appearing in one- and two-loop amplitudes in planar
super-Yang-Mills theory, and . The derivative of
with respect to one of the conformal invariants yields
, while another first-order differential operator applied to
yields . We also introduce some kinematic
variables that rationalize the arguments of the polylogarithms, making it easy
to verify the latter differential equation. We also give a further example of a
six-dimensional integral relevant for amplitudes in
super-Yang-Mills.Comment: 18 pages, 2 figure
One-loop derivation of the Wilson polygon - MHV amplitude duality
We discuss the origin of the Wilson polygon - MHV amplitude duality at the
perturbative level. It is shown that the duality for the MHV amplitudes at
one-loop level can be proven upon the peculiar change of variables in Feynman
parametrization and the use of the relation between Feynman integrals at the
different space-time dimensions. Some generalization of the duality which
implies the insertion of the particular vertex operator at the Wilson triangle
is found for the 3-point function. We discuss analytical structure of Wilson
loop diagrams and present the corresponding Landau equations. The geometrical
interpretation of the loop diagram in terms of the hyperbolic geometry is
discussed.Comment: 29 page
Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
We provide a simple analytic formula for the two-loop six-point ratio
function of planar N = 4 super Yang-Mills theory. This result extends the
analytic knowledge of multi-loop six-point amplitudes beyond those with maximal
helicity violation. We make a natural ansatz for the symbols of the relevant
functions appearing in the two-loop amplitude, and impose various consistency
conditions, including symmetry, the absence of spurious poles, the correct
collinear behaviour, and agreement with the operator product expansion for
light-like (super) Wilson loops. This information reduces the ansatz to a small
number of relatively simple functions. In order to fix these parameters
uniquely, we utilize an explicit representation of the amplitude in terms of
loop integrals that can be evaluated analytically in various kinematic limits.
The final compact analytic result is expressed in terms of classical
polylogarithms, whose arguments are rational functions of the dual conformal
cross-ratios, plus precisely two functions that are not of this type. One of
the functions, the loop integral \Omega^{(2)}, also plays a key role in a new
representation of the remainder function R_6^{(2)} in the maximally helicity
violating sector. Another interesting feature at two loops is the appearance of
a new (parity odd) \times (parity odd) sector of the amplitude, which is absent
at one loop, and which is uniquely determined in a natural way in terms of the
more familiar (parity even) \times (parity even) part. The second
non-polylogarithmic function, the loop integral \tilde{\Omega}^{(2)},
characterizes this sector. Both \Omega^{(2)} and tilde{\Omega}^{(2)} can be
expressed as one-dimensional integrals over classical polylogarithms with
rational arguments.Comment: 51 pages, 4 figures, one auxiliary file with symbols; v2 minor typo
correction
Amplitudes at Weak Coupling as Polytopes in AdS_5
We show that one-loop scalar box functions can be interpreted as volumes of
geodesic tetrahedra embedded in a copy of AdS_5 that has dual conformal
space-time as boundary. When the tetrahedron is space-like, it lies in a
totally geodesic hyperbolic three-space inside AdS_5, with its four vertices on
the boundary. It is a classical result that the volume of such a tetrahedron is
given by the Bloch-Wigner dilogarithm and this agrees with the standard physics
formulae for such box functions. The combinations of box functions that arise
in the n-particle one-loop MHV amplitude in N=4 super Yang-Mills correspond to
the volume of a three-dimensional polytope without boundary, all of whose
vertices are attached to a null polygon (which in other formulations is
interpreted as a Wilson loop) at infinity.Comment: 16 pages, 5 figure
Basics of Generalized Unitarity
We review generalized unitarity as a means for obtaining loop amplitudes from
on-shell tree amplitudes. The method is generally applicable to both
supersymmetric and non-supersymmetric amplitudes, including non-planar
contributions. Here we focus mainly on N=4 Yang-Mills theory, in the context of
on-shell superspaces. Given the need for regularization at loop level, we also
review a six-dimensional helicity-based superspace formalism and its
application to dimensional and massive regularizations. An important feature of
the unitarity method is that it offers a means for carrying over any identified
tree-level property of on-shell amplitudes to loop level, though sometimes in a
modified form. We illustrate this with examples of dual conformal symmetry and
a recently discovered duality between color and kinematics.Comment: 37 pages, 10 figures. Invited review for a special issue of Journal
of Physics A devoted to "Scattering Amplitudes in Gauge Theories", R.
Roiban(ed), M. Spradlin(ed), A. Volovich(ed
Yangian symmetry of light-like Wilson loops
We show that a certain class of light-like Wilson loops exhibits a Yangian
symmetry at one loop, or equivalently, in an Abelian theory. The Wilson loops
we discuss are equivalent to one-loop MHV amplitudes in N=4 super Yang-Mills
theory in a certain kinematical regime. The fact that we find a Yangian
symmetry constraining their functional form can be thought of as the effect of
the original conformal symmetry associated to the scattering amplitudes in the
N=4 theory.Comment: 15 pages, 5 figure
Bootstrapping the three-loop hexagon
We consider the hexagonal Wilson loop dual to the six-point MHV amplitude in
planar N=4 super Yang-Mills theory. We apply constraints from the operator
product expansion in the near-collinear limit to the symbol of the remainder
function at three loops. Using these constraints, and assuming a natural ansatz
for the symbol's entries, we determine the symbol up to just two undetermined
constants. In the multi-Regge limit, both constants drop out from the symbol,
enabling us to make a non-trivial confirmation of the BFKL prediction for the
leading-log approximation. This result provides a strong consistency check of
both our ansatz for the symbol and the duality between Wilson loops and MHV
amplitudes. Furthermore, we predict the form of the full three-loop remainder
function in the multi-Regge limit, beyond the leading-log approximation, up to
a few constants representing terms not detected by the symbol. Our results
confirm an all-loop prediction for the real part of the remainder function in
multi-Regge 3-->3 scattering. In the multi-Regge limit, our result for the
remainder function can be expressed entirely in terms of classical
polylogarithms. For generic six-point kinematics other functions are required.Comment: 36 pages, 1 figure, plus 8 ancillary files containing symbols of
functions; v2 minor typo correction
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