5,069 research outputs found
Using an identity lens : constructive working with children in the criminal justice system
Research has shown that identity, and how you feel about yourself, can be key to moving forward with life and away from crime. Working with the University of Salford, Youth Offending Teams and supported by the Barrow Cadbury Trust, this resource has been developed to promote a constructive, identity-focused approach to ultimately help divert children away from progressing further through the criminal justice system. Using the principles of the Nacro-led Beyond Youth Custody programme, this toolkit outlines how these can be applied to working with children before custody to support them towards positive outcomes and prevent further offending
Towards the Amplituhedron Volume
21 pages; v2: version published in JHEPIt has been recently conjectured that scattering amplitudes in planar N=4 super Yang-Mills are given by the volume of the (dual) amplituhedron. In this paper we show some interesting connections between the tree-level amplituhedron and a special class of differential equations. In particular we demonstrate how the amplituhedron volume for NMHV amplitudes is determined by these differential equations. The new formulation allows for a straightforward geometric description, without any reference to triangulations. Finally we discuss possible implications for volumes related to generic N^kMHV amplitudes.Peer reviewe
On the Classification of Residues of the Grassmannian
We study leading singularities of scattering amplitudes which are obtained as
residues of an integral over a Grassmannian manifold. We recursively do the
transformation from twistors to momentum twistors and obtain an iterative
formula for Yangian invariants that involves a succession of dualized twistor
variables. This turns out to be useful in addressing the problem of classifying
the residues of the Grassmannian. The iterative formula leads naturally to new
coordinates on the Grassmannian in terms of which both composite and
non-composite residues appear on an equal footing. We write down residue
theorems in these new variables and classify the independent residues for some
simple examples. These variables also explicitly exhibit the distinct solutions
one expects to find for a given set of vanishing minors from Schubert calculus.Comment: 20 page
On All-loop Integrands of Scattering Amplitudes in Planar N=4 SYM
We study the relationship between the momentum twistor MHV vertex expansion
of planar amplitudes in N=4 super-Yang-Mills and the all-loop generalization of
the BCFW recursion relations. We demonstrate explicitly in several examples
that the MHV vertex expressions for tree-level amplitudes and loop integrands
satisfy the recursion relations. Furthermore, we introduce a rewriting of the
MHV expansion in terms of sums over non-crossing partitions and show that this
cyclically invariant formula satisfies the recursion relations for all numbers
of legs and all loop orders.Comment: 34 pages, 17 figures; v2: Minor improvements to exposition and
discussion, updated references, typos fixe
Local Spacetime Physics from the Grassmannian
A duality has recently been conjectured between all leading singularities of
n-particle N^(k-2)MHV scattering amplitudes in N=4 SYM and the residues of a
contour integral with a natural measure over the Grassmannian G(k,n). In this
note we show that a simple contour deformation converts the sum of Grassmannian
residues associated with the BCFW expansion of NMHV tree amplitudes to the CSW
expansion of the same amplitude. We propose that for general k the same
deformation yields the (k-2) parameter Risager expansion. We establish this
equivalence for all MHV-bar amplitudes and show that the Risager degrees of
freedom are non-trivially determined by the GL(k-2) "gauge" degrees of freedom
in the Grassmannian. The Risager expansion is known to recursively construct
the CSW expansion for all tree amplitudes, and given that the CSW expansion
follows directly from the (super) Yang-Mills Lagrangian in light-cone gauge,
this contour deformation allows us to directly see the emergence of local
space-time physics from the Grassmannian.Comment: 22 pages, 13 figures; v2: minor updates, typos correcte
The 1/N-expansion, quantum-classical correspondence and nonclassical states generation in dissipative higher-order anharmonic oscillators
We develop a method for the determination of thecdynamics of dissipative
quantum systems in the limit of large number of quanta N, based on the
1/N-expansion of Heidmann et al. [ Opt. Commun. 54, 189 (1985) ] and the
quantum-classical correspondence. Using this method, we find analytically the
dynamics of nonclassical states generation in the higher-order anharmonic
dissipative oscillators for an arbitrary temperature of a reservoir. We show
that the quantum correction to the classical motion increases with time
quadratically up to some maximal value, which is dependent on the degree of
nonlinearity and a damping constant, and then it decreases. Similarities and
differences with the corresponding behavior of the quantum corrections to the
classical motion in the Hamiltonian chaotic systems are discussed. We also
compare our results obtained for some limiting cases with the results obtained
by using other semiclassical tools and discuss the conditions for validity of
our approach.Comment: 15 pages, RevTEX (EPSF-style), 3 figs. Replaced with final version
(stylistic corrections
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
Light-like polygonal Wilson loops in 3d Chern-Simons and ABJM theory
We study light-like polygonal Wilson loops in three-dimensional Chern-Simons
and ABJM theory to two-loop order. For both theories we demonstrate that the
one-loop contribution to these correlators cancels. For pure Chern-Simons, we
find that specific UV divergences arise from diagrams involving two cusps,
implying the loss of finiteness and topological invariance at two-loop order.
Studying those UV divergences we derive anomalous conformal Ward identities for
n-cusped Wilson loops which restrict the finite part of the latter to
conformally invariant functions. We also compute the four-cusp Wilson loop in
ABJM theory to two-loop order and find that the result is remarkably similar to
that of the corresponding Wilson loop in N=4 SYM. Finally, we speculate about
the existence of a Wilson loop/scattering amplitude relation in ABJM theory.Comment: 37 pages, many figures; v2: references added, minor changes; v3:
references added, sign error fixed and note adde
Proof of the MHV vertex expansion for all tree amplitudes in N=4 SYM theory
We prove the MHV vertex expansion for all tree amplitudes of N=4 SYM theory.
The proof uses a shift acting on all external momenta, and we show that every
N^kMHV tree amplitude falls off as 1/z^k, or faster, for large z under this
shift. The MHV vertex expansion allows us to derive compact and efficient
generating functions for all N^kMHV tree amplitudes of the theory. We also
derive an improved form of the anti-NMHV generating function. The proof leads
to a curious set of sum rules for the diagrams of the MHV vertex expansion.Comment: 40 pages, 7 figure
Dualities for Loop Amplitudes of N=6 Chern-Simons Matter Theory
In this paper we study the one- and two-loop corrections to the four-point
amplitude of N=6 Chern-Simons matter theory. Using generalized unitarity
methods we express the one- and two-loop amplitudes in terms of dual-conformal
integrals. Explicit integration by using dimensional reduction gives vanishing
one-loop result as expected, while the two-loop result is non-vanishing and
matches with the Wilson loop computation. Furthermore, the two-loop correction
takes the same form as the one-loop correction to the four-point amplitude of
N=4 super Yang-Mills. We discuss possible higher loop extensions of this
correspondence between the two theories. As a side result, we extend the method
of dimensional reduction for three dimensions to five dimensions where dual
conformal symmetry is most manifest, demonstrating significant simplification
to the computation of integrals.Comment: 32 pages and 6 figures. v2: minus sign corrections, ref updated v3:
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