2,258 research outputs found
Vanishing of 3-Loop Jacobi Diagrams of Odd Degree
We prove the vanishing of the space of 3-loop Jacobi diagrams of odd degree.
This implies that no 3-loop finite-type invariant can distinguish between a
knot and its inverse.Comment: 13 pages. Section on the even degree case expanded. Various minor
correction
New BCJ representations for one-loop amplitudes in gauge theories and gravity
We explain a procedure to manifest the Bern-Carrasco-Johansson duality
between color and kinematics in -point one-loop amplitudes of a variety of
supersymmetric gauge theories. Explicit amplitude representations are
constructed through a systematic reorganization of the integrands in the
Cachazo-He-Yuan formalism. Our construction holds for any nonzero number of
supersymmetries and does not depend on the number of spacetime dimensions. The
cancellations from supersymmetry multiplets in the loop as well as the
resulting power counting of loop momenta is manifested along the lines of the
corresponding superstring computations. The setup is used to derive the
one-loop version of the Kawai-Lewellen-Tye formula for the loop integrands of
gravitational amplitudes.Comment: 58 + 15 page
Topological Field Theory Interpretation of String Topology
The string bracket introduced by Chas and Sullivan [math.GT/9911159] is
reinterpreted from the point of view of topological field theories in the
Batalin-Vilkovisky or BRST formalisms. Namely, topological action functionals
for gauge fields (generalizing Chern-Simons and BF theories) are considered
together with generalized Wilson loops. The latter generate a (Poisson or
Gerstenhaber) algebra of functionals with values in the -equivariant
cohomology of the loop space of the manifold on which the theory is defined. It
is proved that, in the case of with standard representation, the
(Poisson or BV) bracket of two generalized Wilson loops applied to two cycles
is the same as the generalized Wilson loop applied to the string bracket of the
cycles. Generalizations to other groups are briefly described.Comment: 27 pages, 2 figure
BCJ duality and double copy in the closed string sector
This paper is focused on the loop-level understanding of the
Bern-Carrasco-Johansson double copy procedure that relates the integrands of
gauge theory and gravity scattering amplitudes. At four points, the first
non-trivial example of that construction is one-loop amplitudes in N=2
super-Yang-Mills theory and the symmetric realization of N=4 matter-coupled
supergravity. Our approach is to use both field and string theory in parallel
to analyze these amplitudes. The closed string provides a natural framework to
analyze the BCJ construction, in which the left- and right-moving sectors
separately create the color and kinematics at the integrand level. At tree
level, in a five-point example, we show that the Mafra-Schlotterer-Stieberger
procedure gives a new direct proof of the color-kinematics double copy. We
outline the extension of that argument to n points. At loop level, the
field-theoretic BCJ construction of N=2 SYM amplitudes introduces new terms,
unexpected from the string theory perspective. We discuss to what extent we can
relate them to the terms coming from the interactions between left- and
right-movers in the string-theoretic gravity construction.Comment: 46 pages, 8 figures, 2 tables; v3 significantly revised published
versio
On soft singularities at three loops and beyond
We report on further progress in understanding soft singularities of massless
gauge theory scattering amplitudes. Recently, a set of equations was derived
based on Sudakov factorization, constraining the soft anomalous dimension
matrix of multi-leg scattering amplitudes to any loop order, and relating it to
the cusp anomalous dimension. The minimal solution to these equations was shown
to be a sum over color dipoles. Here we explore potential contributions to the
soft anomalous dimension that go beyond the sum-over-dipoles formula. Such
contributions are constrained by factorization and invariance under rescaling
of parton momenta to be functions of conformally invariant cross ratios.
Therefore, they must correlate the color and kinematic degrees of freedom of at
least four hard partons, corresponding to gluon webs that connect four eikonal
lines, which first appear at three loops. We analyze potential contributions,
combining all available constraints, including Bose symmetry, the expected
degree of transcendentality, and the singularity structure in the limit where
two hard partons become collinear. We find that if the kinematic dependence is
solely through products of logarithms of cross ratios, then at three loops
there is a unique function that is consistent with all available constraints.
If polylogarithms are allowed to appear as well, then at least two additional
structures are consistent with the available constraints.Comment: v2: revised version published in JHEP (minor corrections in Sec. 4;
added discussion in Sec. 5.3; refs. added); v3: minor corrections (eqs. 5.11,
5.12 and 5.29); 38 pages, 3 figure
Area versus Length Distribution for Closed Random Walks
Using a connection between the -oscillator algebra and the coefficients of
the high temperature expansion of the frustrated Gaussian spin model, we derive
an exact formula for the number of closed random walks of given length and
area, on a hypercubic lattice, in the limit of infinite number of dimensions.
The formula is investigated in detail, and asymptotic behaviours are evaluated.
The area distribution in the limit of long loops is computed. As a byproduct,
we obtain also an infinite set of new, nontrivial identities.Comment: 17 page
Abelian Z-theory: NLSM amplitudes and alpha'-corrections from the open string
In this paper we derive the tree-level S-matrix of the effective theory of
Goldstone bosons known as the non-linear sigma model (NLSM) from string theory.
This novel connection relies on a recent realization of tree-level
open-superstring S-matrix predictions as a double copy of super-Yang-Mills
theory with Z-theory --- the collection of putative scalar effective field
theories encoding all the alpha'-dependence of the open superstring. Here we
identify the color-ordered amplitudes of the NLSM as the low-energy limit of
abelian Z-theory. This realization also provides natural higher-derivative
corrections to the NLSM amplitudes arising from higher powers of alpha' in the
abelian Z-theory amplitudes, and through double copy also to Born-Infeld and
Volkov-Akulov theories. The Kleiss-Kuijf and Bern-Carrasco-Johansson relations
obeyed by Z-theory amplitudes thereby apply to all alpha'-corrections of the
NLSM. As such we naturally obtain a cubic-graph parameterization for the
abelian Z-theory predictions whose kinematic numerators obey the duality
between color and kinematics to all orders in alpha'.Comment: 37 pages; v2: references, explanations and arguments for
factorization added; published versio
Scattering in Mass-Deformed N>=4 Chern-Simons Models
We investigate the scattering matrix in mass-deformed N>=4 Chern-Simons
models including as special cases the BLG and ABJM theories of multiple M2
branes. Curiously the structure of this scattering matrix in three spacetime
dimensions is equivalent to (a) the two-dimensional worldsheet matrix found in
the context of AdS/CFT integrability and (b) the R-matrix of the
one-dimensional Hubbard model. The underlying reason is that all three models
are based on an extension of the psu(2|2) superalgebra which constrains the
matrix completely. We also compute scattering amplitudes in one-loop field
theory and find perfect agreement with scattering unitarity.Comment: 63 pages, v2: minor corrections, v3: minor improvement
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