18 research outputs found
On tree amplitudes of supersymmetric Einstein-Yang-Mills theory
We present a new formula for all single trace tree amplitudes in four
dimensional super Yang-Mills coupled to Einstein supergravity. Like the
Cachazo-He-Yuan formula, our expression is supported on solutions of the
scattering equations, but with momenta written in terms of spinor helicity
variables. Supersymmetry and parity are both manifest. In the pure gravity and
pure Yang-Mills sectors, it reduces to the known twistor-string formulae. We
show that the formula behaves correctly under factorization and sketch how
these amplitudes may be obtained from a four-dimensional (ambi)twistor string.Comment: 14 pages, no figures. v2: erroneous formulae removed, improved
discussion of factorizatio
Perturbative gravity at null infinity
We describe a theory that lives on the null conformal boundary of
asymptotically flat space-time, and whose states encode the radiative modes of
(super)gravity. We study the induced action of the BMS group, verifying that
the Ward identity for certain BMS supertranslations is equivalent to Weinberg's
soft graviton theorem in the bulk. The subleading behaviour of soft gravitons
may also be obtained from a Ward identity for certain superrotation generators
in the extended BMS algebra proposed by Barnich & Troessaert. We show that the
theory computes the complete classical gravitational S-matrix, perturbatively
around the Minkowski vacuum.Comment: 14 pages, no figures. v2: typos corrected, references adde
Tree-Level Formalism
We review two novel techniques used to calculate tree-level scattering
amplitudes efficiently: MHV diagrams, and on-shell recursion relations. For the
MHV diagrams, we consider applications to tree-level amplitudes and focus in
particular on the N=4 supersymmetric formulation. We also briefly describe the
derivation of loop amplitudes using MHV diagrams. For the recursion relations,
after presenting their general proof, we discuss several applications to
massless theories with and without supersymmetry, to theories with massive
particles, and to graviton amplitudes in General Relativity. This article is an
invited review for a special issue of Journal of Physics A devoted to
"Scattering Amplitudes in Gauge Theories".Comment: 40 pages, 8 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); v2: minor corrections,
references adde
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
Hidden Simplicity of Gauge Theory Amplitudes
These notes were given as lectures at the CERN Winter School on Supergravity,
Strings and Gauge Theory 2010. We describe the structure of scattering
amplitudes in gauge theories, focussing on the maximally supersymmetric theory
to highlight the hidden symmetries which appear. Using the BCFW recursion
relations we solve for the tree-level S-matrix in N=4 super Yang-Mills theory,
and describe how it produces a sum of invariants of a large symmetry algebra.
We review amplitudes in the planar theory beyond tree-level, describing the
connection between amplitudes and Wilson loops, and discuss the implications of
the hidden symmetries.Comment: 46 pages, 15 figures. v2 ref added, typos fixe
Generic multiloop methods and application to N=4 super-Yang-Mills
We review some recent additions to the tool-chest of techniques for finding
compact integrand representations of multiloop gauge-theory amplitudes -
including non-planar contributions - applicable for N=4 super-Yang-Mills in
four and higher dimensions, as well as for theories with less supersymmetry. We
discuss a general organization of amplitudes in terms of purely cubic graphs,
review the method of maximal cuts, as well as some special D-dimensional
recursive cuts, and conclude by describing the efficient organization of
amplitudes resulting from the conjectured duality between color and kinematic
structures on constituent graphs.Comment: 42 pages, 18 figures, invited review for a special issue of Journal
of Physics A devoted to "Scattering Amplitudes in Gauge Theories", v2 minor
corrections, v3 added reference
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
Ambitwistor strings and the scattering equations at one loop
Ambitwistor strings are chiral, infinite tension analogues of conventional
string theory whose target space is the space of complex null geodesics and
whose spectrum consists exclusively of massless states. At genus zero, these
strings underpin the Cachazo-He-Yuan formulae for tree level scattering of
gravitons, gluons and scalars. In this paper we extend these formulae in a
number of directions. Firstly, we consider Ramond sector vertex operators and
construct simple amplitudes involving space-time fermions. These agree with
tree amplitudes in ten dimensional supergravity and super Yang--Mills. We then
show that, after the usual GSO projections, the ambitwistor string partition
function is modular invariant. We consider the scattering equations at genus
one, and calculate one loop scattering amplitudes for NS-NS external states in
the Type II ambitwistor string. We conjecture that these give new
representations of (the integrand of) one loop supergravity amplitudes and we
show that they have the expected behaviour under factorization of the
worldsheet in both non--separating and separating degenerations.Comment: 34 pages, no figures. v2: improvements to discussion, references
update
Worldsheet factorization for twistor-strings
We study the multiparticle factorization properties of two worldsheet
theories which--at tree-level--describe the scattering of massless particles in
four dimensions: the Berkovits-Witten twistor-string for N=4 super-Yang-Mills
coupled to N=4 conformal supergravity, and the Skinner twistor-string for N=8
supergravity. By considering these string-like theories, we can study
factorization at the level of the worldsheet before any Wick contractions or
integrals have been performed; this is much simpler than considering the
factorization properties of the amplitudes themselves. In Skinner's
twistor-string this entails the addition of worldsheet gravity as well as a
formalism that represents all external states in a manifestly symmetric way,
which we develop explicitly at genus zero. We confirm that the scattering
amplitudes of Skinner's theory, as well as the gauge theory amplitudes for the
planar sector of the Berkovits-Witten theory, factorize appropriately at genus
zero. In the non-planar sector, we find behavior indicative of conformal
gravity in the Berkovits-Witten twistor-string. We contrast factorization in
twistor-strings with the story in ordinary string theory, and also make some
remarks on higher genus factorization and disconnected prescriptions.Comment: 50 pages, 7 figures. v2: typos corrected and references update