14,336 research outputs found
The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM
We give an explicit recursive formula for the all L-loop integrand for
scattering amplitudes in N=4 SYM in the planar limit, manifesting the full
Yangian symmetry of the theory. This generalizes the BCFW recursion relation
for tree amplitudes to all loop orders, and extends the Grassmannian duality
for leading singularities to the full amplitude. It also provides a new
physical picture for the meaning of loops, associated with canonical operations
for removing particles in a Yangian-invariant way. Loop amplitudes arise from
the "entangled" removal of pairs of particles, and are naturally presented as
an integral over lines in momentum-twistor space. As expected from manifest
Yangian-invariance, the integrand is given as a sum over non-local terms,
rather than the familiar decomposition in terms of local scalar integrals with
rational coefficients. Knowing the integrands explicitly, it is straightforward
to express them in local forms if desired; this turns out to be done most
naturally using a novel basis of chiral, tensor integrals written in
momentum-twistor space, each of which has unit leading singularities. As simple
illustrative examples, we present a number of new multi-loop results written in
local form, including the 6- and 7-point 2-loop NMHV amplitudes. Very concise
expressions are presented for all 2-loop MHV amplitudes, as well as the 5-point
3-loop MHV amplitude. The structure of the loop integrand strongly suggests
that the integrals yielding the physical amplitudes are "simple", and
determined by IR-anomalies. We briefly comment on extending these ideas to more
general planar theories.Comment: 46 pages; v2: minor changes, references adde
The S-Matrix in Twistor Space
The simplicity and hidden symmetries of (Super) Yang-Mills and (Super)Gravity
scattering amplitudes suggest the existence of a "weak-weak" dual formulation
in which these structures are made manifest at the expense of manifest
locality. We suggest that this dual description lives in (2,2) signature and is
naturally formulated in twistor space. We recast the BCFW recursion relations
in an on-shell form that begs to be transformed into twistor space. Our twistor
transformation is inspired by Witten's, but differs in treating twistor and
dual twistor variables more equally. In these variables the three and
four-point amplitudes are amazingly simple; the BCFW relations are represented
by diagrammatic rules that precisely define the "twistor diagrams" of Andrew
Hodges. The "Hodges diagrams" for Yang-Mills theory are disks and not trees;
they reveal striking connections between amplitudes and suggest a new form for
them in momentum space. We also obtain a twistorial formulation of gravity. All
tree amplitudes can be combined into an "S-Matrix" functional which is the
natural holographic observable in asymptotically flat space; the BCFW formula
turns into a quadratic equation for this "S-Matrix", providing a holographic
description of N=4 SYM and N=8 Supergravity at tree level. We explore loop
amplitudes in (2,2) signature and twistor space, beginning with a discussion of
IR behavior. We find that the natural pole prescription renders the amplitudes
well-defined and free of IR divergences. Loop amplitudes vanish for generic
momenta, and in twistor space are even simpler than their tree-level
counterparts! This further supports the idea that there exists a sharply
defined object corresponding to the S-Matrix in (2,2) signature, computed by a
dual theory naturally living in twistor space.Comment: V1: 46 pages + 23 figures. Less telegraphic abstract in the body of
the paper. V2: 49 pages + 24 figures. Largely expanded set of references
included. Some diagrammatic clarifications added, minor typo fixe
Wilson lines in the MHV action
The MHV action is the Yang-Mills action quantized on the light-front, where
the two explicit physical gluonic degrees of freedom have been canonically
transformed to a new set of fields. This transformation leads to the action
with vertices being off-shell continuations of the MHV amplitudes. We show that
the solution to the field transformation expressing one of the new fields in
terms of the Yang-Mills field is a certain type of the Wilson line. More
precisely, it is a straight infinite gauge link with a slope extending to the
light-cone minus and the transverse direction. One of the consequences of that
fact is that certain MHV vertices reduced partially on-shell are gauge
invariant -- a fact discovered before using conventional light-front
perturbation theory. We also analyze the diagrammatic content of the field
transformations leading to the MHV action. We found that the diagrams for the
solution to the transformation (given by the Wilson line) and its inverse
differ only by light-front energy denominators. Further, we investigate the
coordinate space version of the inverse solution to the one given by the Wilson
line. We find an explicit expression given by a power series in fields. We also
give a geometric interpretation to it by means of a specially defined vector
field. Finally, we discuss the fact that the Wilson line solution to the
transformation is directly related to the all-like helicity gluon wave
function, while the inverse functional is a generating functional for solutions
of self-dual Yang-Mills equations.Comment: 44 pages, a few figure
Hermitian vs. Anti-Hermitian 1-Matrix Models and Their Hierarchies
Building on a recent work of \v C. Crnkovi\'c, M. Douglas and G. Moore, a
study of multi-critical multi-cut one-matrix models and their associated
integrable hierarchies, is further pursued. The double scaling limits
of hermitian matrix models with different scaling ans\"atze, lead, to the KdV
hierarchy, to the modified KdV hierarchy and part of the non-linear
Schr\"odinger hierarchy. Instead, the anti-hermitian matrix model, in the
two-arc sector, results in the Zakharov-Shabat hierarchy, which contains both
KdV and mKdV as reductions. For all the hierarchies, it is found that the
Virasoro constraints act on the associated tau-functions. Whereas it is known
that the ZS and KdV models lead to the Virasoro constraints of an
vacuum, we find that the mKdV model leads to the Virasoro constraints of a
highest weight state with arbitrary conformal dimension.Comment: 31 page
RNA Folding and Large N Matrix Theory
We formulate the RNA folding problem as an matrix field theory.
This matrix formalism allows us to give a systematic classification of the
terms in the partition function according to their topological character. The
theory is set up in such a way that the limit yields the
so-called secondary structure (Hartree theory). Tertiary structure and
pseudo-knots are obtained by calculating the corrections to the
partition function. We propose a generalization of the Hartree recursion
relation to generate the tertiary structure.Comment: 29 pages (LaTex), 13 figures (eps). Missing paragraph and figure
adde
Topological expansion of beta-ensemble model and quantum algebraic geometry in the sectorwise approach
We solve the loop equations of the -ensemble model analogously to the
solution found for the Hermitian matrices . For \beta=1y^2=U(x)\beta((\hbar\partial)^2-U(x))\psi(x)=0\hbar\propto
(\sqrt\beta-1/\sqrt\beta)/Ny^2-U(x)[y,x]=\hbarF_h-expansion at arbitrary . The set of "flat"
coordinates comprises the potential times and the occupation numbers
\widetilde{\epsilon}_\alpha\mathcal F_0\widetilde{\epsilon}_\alpha$.Comment: 58 pages, 7 figure
Adaptive Mesh Refinement for Characteristic Grids
I consider techniques for Berger-Oliger adaptive mesh refinement (AMR) when
numerically solving partial differential equations with wave-like solutions,
using characteristic (double-null) grids. Such AMR algorithms are naturally
recursive, and the best-known past Berger-Oliger characteristic AMR algorithm,
that of Pretorius & Lehner (J. Comp. Phys. 198 (2004), 10), recurses on
individual "diamond" characteristic grid cells. This leads to the use of
fine-grained memory management, with individual grid cells kept in
2-dimensional linked lists at each refinement level. This complicates the
implementation and adds overhead in both space and time.
Here I describe a Berger-Oliger characteristic AMR algorithm which instead
recurses on null \emph{slices}. This algorithm is very similar to the usual
Cauchy Berger-Oliger algorithm, and uses relatively coarse-grained memory
management, allowing entire null slices to be stored in contiguous arrays in
memory. The algorithm is very efficient in both space and time.
I describe discretizations yielding both 2nd and 4th order global accuracy.
My code implementing the algorithm described here is included in the electronic
supplementary materials accompanying this paper, and is freely available to
other researchers under the terms of the GNU general public license.Comment: 37 pages, 15 figures (40 eps figure files, 8 of them color; all are
viewable ok in black-and-white), 1 mpeg movie, uses Springer-Verlag svjour3
document class, includes C++ source code. Changes from v1: revised in
response to referee comments: many references added, new figure added to
better explain the algorithm, other small changes, C++ code updated to latest
versio
Recursive numerical calculus of one-loop tensor integrals
A numerical approach to compute tensor integrals in one-loop calculations is
presented. The algorithm is based on a recursion relation which allows to
express high rank tensor integrals as a function of lower rank ones. At each
level of iteration only inverse square roots of Gram determinants appear. For
the phase-space regions where Gram determinants are so small that numerical
problems are expected, we give general prescriptions on how to construct
reliable approximations to the exact result without performing Taylor
expansions. Working in 4+epsilon dimensions does not require an analytic
separation of ultraviolet and infrared/collinear divergences, and, apart from
trivial integrals that we compute explicitly, no additional ones besides the
standard set of scalar one-loop integrals are needed.Comment: Typo corrected in formula 79. 22 pages, Latex, 1 figure, uses
axodraw.st
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