Some analytic results for two-loop scattering amplitudes

Luis F. Alday

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Some analytic results for two-loop scattering amplitudes

Luis F. Alday

High Energy Physics - Theory · Submitted 6 Sep 2010

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The authors' main claim We present analytic results for the finite diagrams contributing to the two-loop eight-point MHV scattering amplitude of planar N=4 SYM.

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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.

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present analytic results for the finite diagrams contributing to the two-loop eight-point MHV scattering amplitude of planar N=4 SYM.	contribution 0.75	0.96	0.79	asserted 1.00
s1	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.	method 0.96	0.86	0.07	asserted 0.99
s2	The theory of motives can be used to vastly simplify the results, which can be expressed as simple combinations of classical polylogarithms.	method 0.48	0.63	0.31	asserted 0.74

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The paper

Original abstract

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.

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References

Forward citations

Cited by 6 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Scattering Amplitudes and the Positive Grassmannian
hep-th 2012-12 accept novelty 8.0 of 10

Scattering amplitudes in planar N=4 super Yang-Mills are organized by on-shell diagrams whose invariant content is a cell of the positive Grassmannian, labeled by a permutation.
Analytic result for the two-loop six-point NMHV amplitude in N=4 super Yang-Mills theory
hep-th 2011-11 accept novelty 7.0 of 10

The authors derive the first exact analytic expression for the two-loop six-point NMHV scattering amplitude in planar N=4 super Yang-Mills theory, written in classical polylogarithms plus two new special functions.
Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills
hep-th 2011-05 conditional novelty 7.0 of 10

A conjectured superspace generalization of scattering amplitudes yields, via exact dual superconformal Ward identities, the total differential and symbol of the two-loop MHV remainder function for any number of points.
The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space
hep-th 2010-09 conditional novelty 7.0 of 10

The full planar S-matrix of N=4 SYM is conjectured to equal a supersymmetric twistor-space Wilson loop correlator, supported by matching tree and loop-integrand results.
Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes
hep-ph 2012-03 conditional novelty 6.0 of 10

The coproduct of multiple polylogarithms, unlike the symbol, retains zeta-value information, and it is used to compress the two-loop Higgs-plus-three-gluon helicity amplitudes into classical polylogarithms.

Show 1 more citing papers

From polygons and symbols to polylogarithmic functions
math-ph 2011-10 conditional novelty 5.0 of 10

The paper gives a polygon-based recipe for the symbol of a multiple polylogarithm, an algorithm for integrating symbols back to functions, and a spanning set for harmonic polylogarithms up to weight four.

How the literature uses it

17 arXiv papers cite this in 26 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 14 sentences without a confident label

Builds on it 2 papers

The same equations can be used for higher loops, extending the results of [ref] .

Differential equations for multi-loop integrals and two-dimensional kinematics · Introduction · 1.00
Subsequently, we can further restrict the $4d$ momentum twistors to a subspace of $2d$ kinematics which can be very simply parameterized using $6$ independent cross-ratios, as a generalization of the parameterization used in [ref] : [equation] In terms of these variables one may compute the $x_ij^2 = (Z_2i-1 Z_2i Z_2j-1 Z_2j)$, so that the $9$ cross-ratios [ref] are given by u_1 = χ_1^- (χ_3^-+1) (χ_1^+-χ_3^+)/(χ_1^-+1) χ_3^- (χ_1^+-χ_2^+), u_2 = (χ_1^-+1) (χ_2^++1)/(χ_3^-+1) (χ_3^++1), u_3 = (χ_2^–χ_3^-) χ_2^+ (χ_3^++1)/(χ_2^–χ_1^-) (χ_2^++1) χ_3^+, u_4 = χ_3^++1/χ_1^+ χ_1^-+χ_1^-+χ_1^++1, u_5 = (χ_2^–χ_1^-) (χ_1^++1) χ_3^+/χ_2^- χ_1^+ (χ_3^++1), u_6 = χ_2^- (χ_1^–χ_3^-) χ_1^+ (χ_2^+-χ_3^+)/(χ_2^–χ_1^-) χ_3^- (χ_1^+-χ_2^+) χ_3^+, u_7 = χ_2^- χ_1^+/(χ_2^-+1) (χ_1^++1), u_8 = (χ_2^-+1) χ_3^- (χ_1^+-χ_2^+)/χ_2^- (χ_3^-+1) χ_1^+, u_9 = χ_3^-+1/χ_2^+ χ_2^-+χ_2^-+χ_2^++1 .

Star Integrals, Convolutions and Simplices · Details on the double box computation · 0.87

Uses its method or tool 2 papers

[ref] we can distinguish the first sum from the second by projecting the respective symbols onto their symmetric or alternating parts: each term in the second sum will give zero contribution for the latter one, while each summand in the first sum will give a non-zero contribution.

From polygons and symbols to polylogarithmic functions · A simple example · 1.00
The six-dimensional notation offfers a particularly efficient way to achieve this; let us go through the various step in detail (see also [ref] ).

Superconformal symmetry and two-loop amplitudes in planar N=4 super Yang-Mills · Contributions $C^(2)$ and $C^(3)$ · 0.88

Compares with it 1 paper

We finally find that [equation] This symbol corresponds to the one found in [ref] when appropriate differential operator and change of variables are taken into account.

Differential equations for multi-loop integrals and two-dimensional kinematics · Double-pentagon integral $F^(2)_c$ · 1.00

Cites it as background 6 papers

The two-loop remainder function $R^(2)_8$ for the scattering of eight gluons was calculated by Del Duca, Duhr and Smirnov [ref] and its diagrammatic structure was analyzed by Alday [ref] .

BFKL approach and six-particle MHV amplitude in N=4 super Yang-Mills · Introduction · 0.97
Since their introduction, symbols have seen many applications in physics [ref] .

Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes · Introduction · 0.83
Over the last couple of years, twistor techniques have been tremendously successful in describing scattering amplitudes of $N=4$ SYM at weak coupling, both at the level of the integrand, see e.g. [ref] , and for obtaining analytic integrated expressions, [ref] .

Higher loop mixed correlators in N=4 SYM · Introduction and main results · 0.98

What it cites · 32 sentences without a confident label

Uses the method or tool of 4 papers

We used the representation of [ref] , which gives such amplitude as certain integral over twistorial variables.

The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM · Conclusions · 0.97
The method is based on the symmetries of the symbol (see [ref] ).

Classical Polylogarithms for Amplitudes and Wilson Loops · Diagram of type $(b)$ · 0.98
now we are on position of performing to integral over $Z^AB$, see for instance [ref] .

The box integrals in momentum-twistor geometry · One loop example · 0.84

Cites as background 9 papers

Such efforts led to a proposal for an explicit recursive formula for the all loop integrand for scattering amplitudes in planar $N=4$ SYM [ref] [See also [ref] .].

On BCFW shifts of integrands and integrals · Introduction · 0.87
Such efforts led to a proposal for an explicit recursive formula for the all loop integrand for scattering amplitudes in planar $N=4$ SYM [ref] [See also [ref] .].

The All-Loop Integrand For Scattering Amplitudes in Planar N=4 SYM · Introduction · 0.87
In particular, in [ref] , an expression in terms of integrals for two-loop Wilson loops with any number of edges was given, making their numerical computation feasible.

Two-Loop Polygon Wilson Loops in N=4 SYM · Introduction · 0.81

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