104,649 research outputs found
Towards the n-point one-loop superstring amplitude III: One-loop correlators and their double-copy structure
In this final part of a series of three papers, we will assemble
supersymmetric expressions for one-loop correlators in pure-spinor superspace
that are BRST invariant, local, and single valued. A key driving force in this
construction is the generalization of a so far unnoticed property at tree
level; the correlators have the symmetry structure akin to Lie polynomials.
One-loop correlators up to seven points are presented in a variety of
representations manifesting different subsets of their defining properties.
These expressions are related via identities obeyed by the kinematic
superfields and worldsheet functions spelled out in the first two parts of this
series and reflecting a duality between the two kinds of ingredients.
Interestingly, the expression for the eight-point correlator following from
our method seems to capture correctly all the dependence on the worldsheet
punctures but leaves undetermined the coefficient of the holomorphic Eisenstein
series . By virtue of chiral splitting, closed-string correlators
follow from the double copy of the open-string results.Comment: 77 pages, v2: published versio
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
A minimal approach to the scattering of physical massless bosons
Tree and loop level scattering amplitudes which involve physical massless
bosons are derived directly from physical constraints such as locality,
symmetry and unitarity, bypassing path integral constructions. Amplitudes can
be projected onto a minimal basis of kinematic factors through linear algebra,
by employing four dimensional spinor helicity methods or at its most general
using projection techniques. The linear algebra analysis is closely related to
amplitude relations, especially the Bern-Carrasco-Johansson relations for gluon
amplitudes and the Kawai-Lewellen-Tye relations between gluons and graviton
amplitudes. Projection techniques are known to reduce the computation of loop
amplitudes with spinning particles to scalar integrals. Unitarity, locality and
integration-by-parts identities can then be used to fix complete tree and loop
amplitudes efficiently. The loop amplitudes follow algorithmically from the
trees. A range of proof-of-concept examples is presented. These include the
planar four point two-loop amplitude in pure Yang-Mills theory as well as a
range of one loop amplitudes with internal and external scalars, gluons and
gravitons. Several interesting features of the results are highlighted, such as
the vanishing of certain basis coefficients for gluon and graviton amplitudes.
Effective field theories are naturally and efficiently included into the
framework. The presented methods appear most powerful in non-supersymmetric
theories in cases with relatively few legs, but with potentially many loops.
For instance, iterated unitarity cuts of four point amplitudes for
non-supersymmetric gauge and gravity theories can be computed by matrix
multiplication, generalising the so-called rung-rule of maximally
supersymmetric theories. The philosophy of the approach to kinematics also
leads to a technique to control color quantum numbers of scattering amplitudes
with matter.Comment: 65 pages, exposition improved, typos correcte
Scattering equations and virtuous kinematic numerators and dual-trace functions
Inspired by recent developments on scattering equations, we present a
constructive procedure for computing symmetric, amplitude-encoded, BCJ
numerators for n-point gauge-theory amplitudes, thus satisfying the three
virtues identified by Broedel and Carrasco. We also develop a constructive
procedure for computing symmetric, amplitude-encoded dual-trace functions (tau)
for n-point amplitudes. These can be used to obtain symmetric kinematic
numerators that automatically satisfy color-kinematic duality. The S_n symmetry
of n-point gravity amplitudes formed from these symmetric dual-trace functions
is completely manifest. Explicit expressions for four- and five-point
amplitudes are presented.Comment: 24 pages; v2: minor sign corrections, added references; v3: minor
corrections, published versio
On Three-Algebra and Bi-Fundamental Matter Amplitudes and Integrability of Supergravity
We explore tree-level amplitude relations for SU(N)xSU(M) bi-fundamental
matter theories. Embedding the group-theory structure in a Lie three-algebra,
we derive Kleiss-Kuijf-like relations for bi-fundamental matter theories in
general dimension. We investigate the three-algebra color-kinematics duality
for these theories. Unlike the Yang-Mills two-algebra case, the three-algebra
Bern-Carrasco-Johansson relations depend on the spacetime dimension and on the
detailed symmetry properties of the structure constants. We find the presence
of such relations in three and two dimensions, and absence in D>3.
Surprisingly, beyond six point, such relations are absent in the
Aharony-Bergman-Jafferis-Maldacena theory for general gauge group, while the
Bagger-Lambert-Gustavsson theory, and its supersymmetry truncations, obey the
color-kinematics duality like clockwork. At four and six points the relevant
partial amplitudes of the two theories are bijectively related, explaining
previous results in the literature. In D=2 the color-kinematics duality gives
results consistent with integrability of two-dimensional
supergravity: The four-point amplitude satisfies a Yang-Baxter equation; the
six- and eight-point amplitudes vanish for certain kinematics away from
factorization channels, as expected from integrability.Comment: 52 page
Double-Copy Structure of One-Loop Open-String Amplitudes
In this Letter, we provide evidence for a new double-copy structure in
one-loop amplitudes of the open superstring. Their integrands with respect to
the moduli space of genus-one surfaces are cast into a form where
gauge-invariant kinematic factors and certain functions of the punctures --
so-called generalized elliptic integrands -- enter on completely symmetric
footing. In particular, replacing the generalized elliptic integrands by a
second copy of kinematic factors maps one-loop open-string correlators to
gravitational matrix elements of the higher-curvature operator R^4.Comment: 5 pages, v2: modifications in the structure to match published
versio
On paths-based criteria for polynomial time complexity in proof-nets
Girard's Light linear logic (LLL) characterized polynomial time in the
proof-as-program paradigm with a bound on cut elimination. This logic relied on
a stratification principle and a "one-door" principle which were generalized
later respectively in the systems L^4 and L^3a. Each system was brought with
its own complex proof of Ptime soundness.
In this paper we propose a broad sufficient criterion for Ptime soundness for
linear logic subsystems, based on the study of paths inside the proof-nets,
which factorizes proofs of soundness of existing systems and may be used for
future systems. As an additional gain, our bound stands for any reduction
strategy whereas most bounds in the literature only stand for a particular
strategy.Comment: Long version of a conference pape
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