1,242 research outputs found
Symbols of One-Loop Integrals From Mixed Tate Motives
We use a result on mixed Tate motives due to Goncharov
(arXiv:alg-geom/9601021) to show that the symbol of an arbitrary one-loop
2m-gon integral in 2m dimensions may be read off directly from its Feynman
parameterization. The algorithm proceeds via recursion in m seeded by the
well-known box integrals in four dimensions. As a simple application of this
method we write down the symbol of a three-mass hexagon integral in six
dimensions.Comment: 13 pages, v2: minor typos correcte
Mellin Amplitudes for Dual Conformal Integrals
Motivated by recent work on the utility of Mellin space for representing
conformal correlators in /CFT, we study its suitability for representing
dual conformal integrals of the type which appear in perturbative scattering
amplitudes in super-Yang-Mills theory. We discuss Feynman-like rules for
writing Mellin amplitudes for a large class of integrals in any dimension, and
find explicit representations for several familiar toy integrals. However we
show that the power of Mellin space is that it provides simple representations
even for fully massive integrals, which except for the single case of the
4-mass box have not yet been computed by any available technology. Mellin space
is also useful for exhibiting differential relations between various multi-loop
integrals, and we show that certain higher-loop integrals may be written as
integral operators acting on the fully massive scalar -gon in
dimensions, whose Mellin amplitude is exactly 1. Our chief example is a very
simple formula expressing the 6-mass double box as a single integral of the
6-mass scalar hexagon in 6 dimensions.Comment: 29+7 page
Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
We show how the Hopf algebra structure of multiple polylogarithms can be used
to simplify complicated expressions for multi-loop amplitudes in perturbative
quantum field theory and we argue that, unlike the recently popularized
symbol-based approach, the coproduct incorporates information about the zeta
values. We illustrate our approach by rewriting the two-loop helicity
amplitudes for a Higgs boson plus three gluons in a simplified and compact form
involving only classical polylogarithms.Comment: 46 page
The last of the simple remainders
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Single-valued harmonic polylogarithms and the multi-Regge limit
We argue that the natural functions for describing the multi-Regge limit of
six-gluon scattering in planar N=4 super Yang-Mills theory are the
single-valued harmonic polylogarithmic functions introduced by Brown. These
functions depend on a single complex variable and its conjugate, (w,w*). Using
these functions, and formulas due to Fadin, Lipatov and Prygarin, we determine
the six-gluon MHV remainder function in the leading-logarithmic approximation
(LLA) in this limit through ten loops, and the next-to-LLA (NLLA) terms through
nine loops. In separate work, we have determined the symbol of the four-loop
remainder function for general kinematics, up to 113 constants. Taking its
multi-Regge limit and matching to our four-loop LLA and NLLA results, we fix
all but one of the constants that survive in this limit. The multi-Regge limit
factorizes in the variables (\nu,n) which are related to (w,w*) by a
Fourier-Mellin transform. We can transform the single-valued harmonic
polylogarithms to functions of (\nu,n) that incorporate harmonic sums,
systematically through transcendental weight six. Combining this information
with the four-loop results, we determine the eigenvalues of the BFKL kernel in
the adjoint representation to NNLLA accuracy, and the MHV product of impact
factors to NNNLLA accuracy, up to constants representing beyond-the-symbol
terms and the one symbol-level constant. Remarkably, only derivatives of the
polygamma function enter these results. Finally, the LLA approximation to the
six-gluon NMHV amplitude is evaluated through ten loops.Comment: 71 pages, 2 figures, plus 10 ancillary files containing analytic
expressions in Mathematica format. V2: Typos corrected and references added.
V3: Typos corrected; assumption about single-Reggeon exchange made explici
The One-Loop One-Mass Hexagon Integral in D=6 Dimensions
We evaluate analytically the one-loop one-mass hexagon in six dimensions. The
result is given in terms of standard polylogarithms of uniform transcendental
weight three.Comment: 9 page
From polygons and symbols to polylogarithmic functions
We present a review of the symbol map, a mathematical tool that can be useful
in simplifying expressions among multiple polylogarithms, and recall its main
properties. A recipe is given for how to obtain the symbol of a multiple
polylogarithm in terms of the combinatorial properties of an associated rooted
decorated polygon. We also outline a systematic approach to constructing a
function corresponding to a given symbol, and illustrate it in the particular
case of harmonic polylogarithms up to weight four. Furthermore, part of the
ambiguity of this process is highlighted by exhibiting a family of non-trivial
elements in the kernel of the symbol map for arbitrary weight.Comment: 75 pages. Mathematica files with the expression of all HPLs up to
weight 4 in terms of the spanning set are include
Collinear and Soft Limits of Multi-Loop Integrands in N=4 Yang-Mills
It has been argued in arXiv:1112.6432 that the planar four-point integrand in
N=4 super Yang-Mills theory is uniquely determined by dual conformal invariance
together with the absence of a double pole in the integrand of the logarithm in
the limit as a loop integration variable becomes collinear with an external
momentum. In this paper we reformulate this condition in a simple way in terms
of the amplitude itself, rather than its logarithm, and verify that it holds
for two- and three-loop MHV integrands for n>4. We investigate the extent to
which this collinear constraint and a constraint on the soft behavior of
integrands can be used to determine integrands. We find an interesting
complementarity whereby the soft constraint becomes stronger while the
collinear constraint becomes weaker at larger n. For certain reasonable choices
of basis at two and three loops the two constraints in unison appear strong
enough to determine MHV integrands uniquely for all n.Comment: 27 pages, 14 figures; v2: very minor change
Some analytic results for two-loop scattering amplitudes
We present analytic results for the finite diagrams contributing to the
two-loop eight-point MHV scattering amplitude of planar N=4 SYM. We use a
recently proposed representation for the integrand of the amplitude in terms of
(momentum) twistors and focus on a restricted kinematics in which the answer
depends only on two independent cross-ratios. The theory of motives can be used
to vastly simplify the results, which can be expressed as simple combinations
of classical polylogarithms.Comment: 18 page
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