89 research outputs found
A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
We define the rigidity of a Feynman integral to be the smallest dimension
over which it is non-polylogarithmic. We argue that massless Feynman integrals
in four dimensions have a rigidity bounded by 2(L-1) at L loops, and we show
that this bound may be saturated for integrals that we call marginal: those
with (L+1)D/2 propagators in (even) D dimensions. We show that marginal Feynman
integrals in D dimensions generically involve Calabi-Yau geometries, and we
give examples of finite four-dimensional Feynman integrals in massless
theory that saturate our predicted bound in rigidity at all loop orders.Comment: 5+2 pages, 11 figures, infinite zoo of Calabi-Yau manifolds. v2
reflects minor changes made for publication. This version is authoritativ
The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
We derive an analytic representation of the ten-particle, two-loop double-box
integral as an elliptic integral over weight-three polylogarithms. To obtain
this form, we first derive a four-fold, rational (Feynman-)parametric
representation for the integral, expressed directly in terms of
dual-conformally invariant cross-ratios; from this, the desired form is easily
obtained. The essential features of this integral are illustrated by means of a
simplified toy model, and we attach the relevant expressions for both integrals
in ancillary files. We propose a normalization for such integrals that renders
all of their polylogarithmic degenerations pure, and we discuss the need for a
new 'symbology' of iterated elliptic/polylogarithmic integrals in order to
bring them to a more canonical form.Comment: 4+2 pages, 2 figures. Explicit results are included as ancillary
files. v2: minor changes made for clarification; references adde
Manifesting Color-Kinematics Duality in the Scattering Equation Formalism
We prove that the scattering equation formalism for Yang-Mills amplitudes can
be used to make manifest the theory's color-kinematics duality. This is
achieved through a concrete reduction algorithm which renders this duality
manifest term-by-term. The reduction follows from the recently derived set of
identities for amplitudes expressed in the scattering equation formalism that
are analogous to monodromy relations in string theory. A byproduct of our
algorithm is a generalization of the identities among gravity and Yang-Mills
amplitudes.Comment: 20 pages, 20 figure
Analytic Representations of Yang-Mills Amplitudes
Scattering amplitudes in Yang-Mills theory can be represented in the
formalism of Cachazo, He and Yuan (CHY) as integrals over an auxiliary
projective space---fully localized on the support of the scattering equations.
Because solving the scattering equations is difficult and summing over the
solutions algebraically complex, a method of directly integrating the terms
that appear in this representation has long been sought. We solve this
important open problem by first rewriting the terms in a manifestly
Mobius-invariant form and then using monodromy relations (inspired by analogy
to string theory) to decompose terms into those for which combinatorial rules
of integration are known. The result is a systematic procedure to obtain
analytic, covariant forms of Yang-Mills tree-amplitudes for any number of
external legs and in any number of dimensions. As examples, we provide compact
analytic expressions for amplitudes involving up to six gluons of arbitrary
helicities.Comment: 29 pages, 43 figures; also included is a Mathematica notebook with
explicit formulae. v2: citations added, and several (important) typos fixe
Scattering Equations and Feynman Diagrams
We show a direct matching between individual Feynman diagrams and integration
measures in the scattering equation formalism of Cachazo, He and Yuan. The
connection is most easily explained in terms of triangular graphs associated
with planar Feynman diagrams in -theory. We also discuss the
generalization to general scalar field theories with interactions,
corresponding to polygonal graphs involving vertices of order . Finally, we
describe how the same graph-theoretic language can be used to provide the
precise link between individual Feynman diagrams and string theory integrands.Comment: 18 pages, 57 figure
Integration Rules for Scattering Equations
As described by Cachazo, He and Yuan, scattering amplitudes in many quantum
field theories can be represented as integrals that are fully localized on
solutions to the so-called scattering equations. Because the number of
solutions to the scattering equations grows quite rapidly, the contour of
integration involves contributions from many isolated components. In this
paper, we provide a simple, combinatorial rule that immediately provides the
result of integration against the scattering equation constraints for any
M\"obius-invariant integrand involving only simple poles. These rules have a
simple diagrammatic interpretation that makes the evaluation of any such
integrand immediate. Finally, we explain how these rules are related to the
computation of amplitudes in the field theory limit of string theory.Comment: 30 pages, 29 figure
Integration Rules for Loop Scattering Equations
We formulate new integration rules for one-loop scattering equations
analogous to those at tree-level, and test them in a number of non-trivial
cases for amplitudes in scalar -theory. This formalism greatly
facilitates the evaluation of amplitudes in the CHY representation at one-loop
order, without the need to explicitly sum over the solutions to the loop-level
scattering equations.Comment: 22 pages, 17 figure
Yangian Symmetry for the Tree Amplituhedron
17 pages, 4 figures; v2: extended discussion of results, minor typos corrected, version published in Journal of Physics ATree-level scattering amplitudes in planar N=4 super Yang-Mills are known to be Yangian-invariant. It has been shown that integrability allows to obtain a general, explicit method to find such invariants. The uplifting of this result to the amplituhedron construction has been an important open problem. In this paper, with the help of methods proper to integrable theories, we successfully fill this gap and clarify the meaning of Yangian invariance for the tree-level amplituhedron. In particular, we construct amplituhedron volume forms from an underlying spin chain. As a by-product of this construction, we also propose a novel on-shell diagrammatics for the amplituhedron.Peer reviewe
New Representations of the Perturbative S-Matrix
We propose a new framework to represent the perturbative S-matrix which is
well-defined for all quantum field theories of massless particles, constructed
from tree-level amplitudes and integrable term-by-term. This representation is
derived from the Feynman expansion through a series of partial fraction
identities, discarding terms that vanish upon integration. Loop integrands are
expressed in terms of "Q-cuts" that involve both off-shell and on-shell
loop-momenta, defined with a precise contour prescription that can be evaluated
by ordinary methods. This framework implies recent results found in the
scattering equation formalism at one-loop, and it has a natural extension to
all orders---even non-planar theories without well-defined forward limits or
good ultraviolet behavior.Comment: 4+1 pages, 4 figure
Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms
We describe a family of finite, four-dimensional, L-loop Feynman integrals that involve weight-(L+1) hyperlogarithms integrated over (L−1)-dimensional elliptically fibered varieties we conjecture to be Calabi-Yau manifolds. At three loops, we identify the relevant K3 explicitly and we provide strong evidence that the four-loop integral involves a Calabi-Yau threefold. These integrals are necessary for the representation of amplitudes in many theories—from massless φ4 theory to integrable theories including maximally supersymmetric Yang-Mills theory in the planar limit—a fact we demonstrate
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